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Functional BvM limits and credible-set calibration

1 · Prerequisites

2 · Summary

Objective

This page develops scalar functional BvM limits and credible-interval calibration in Gaussian white noise, including the nonlinear delta-method condition. It defines the multiscale coefficient space and the type-S and type-H density prior interfaces, then proves the source-specific density weak BvM theorems and their CDF and multiscale credible-band consequences, with the required sampling-process and Gaussian norm-quantile limits. It records the exact smoothness, positivity, topology, centering, and rate assumptions used in each result.

For type S, local quotient-likelihood contraction, wavelet composition and projection estimates, and posterior Laplace bounds prove the weak BvM under both approved coefficient-prior regimes. For type H, the Dirichlet posterior and normalized-Gamma concentration give a direct conjugate proof. The resulting empirical-center credible bands have asymptotically exact coverage; the stated additional coefficient smoothness and undersmoothing assumptions give the diameter bound.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Little-o multiscale coefficient space M0(w)

Definition

Use the wavelet coefficient levels $(l, k)$ and multiplicities $∣ K_{l} ∣ = 2^{l}$ from Logarithmic negative Sobolev spaces for white noise. The same coefficient-space definition applies to the Haar system used for type-H histograms: detail levels $l \geq 0$ have $2^{l}$ coefficients, and the single scaling coefficient is an additional finite coarsest coordinate, included in the norm with weight $1$ . Finite projections always retain that scaling coordinate; this finite convention does not alter any tail assertion. Let $w = (w_{l})$ be nondecreasing with $w_{l} \geq 1$ . Define $M (w) = {x = (x_{l k}) : ∥ x ∥_{M (w)} : = \sup_{l} \frac{\max_{k \in K_{l}} ∣ x_{l k} ∣}{w_{l}} < \infty}$ and its closed little-o subspace $M_{0} (w) = {x \in M (w) : \frac{\max_{k \in K_{l}} ∣ x_{l k} ∣}{w_{l}} ⟶ 0 as l \to \infty},$ with the same norm. Call $w$ admissible when $w_{l} / \sqrt{l} \to \infty$ . The finite coarsest level is included in the norm and does not affect this limit. Elements are coefficient sequences; in particular, they need not be $L^{2}$ functions. The multiscale coefficient space is separable and has vanishing projection tails ↗ proves that $M_{0} (w)$ is a separable Banach space, that $L^{2}$ coefficient sequences embed in it, and that finite-level projections converge in its norm.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Smooth linear functionals on the white-noise topology

Definition

Fix the real orthonormal wavelet basis and level weights $a_{l}$ of Logarithmic negative Sobolev spaces for white noise. For $0 \leq s < S$ , define the basis Sobolev space $H_{2}^{s} = {g \in L^{2} ([0, 1]) : \sum_{l, k} a_{l}^{2 s} ∣ ⟨ g, ψ_{l k} ⟩ ∣^{2} < \infty} .$ For $g \in H_{2}^{s}$ , $s > 1 / 2$ , and $f \in L^{2} ([0, 1])$ , define $L_{g} (f) = ⟨ f, g ⟩_{L^{2}}$ . For $x = (x_{l k}) \in H (δ)$ , the intended extension is the coefficient pairing $L_{g} (x) = \sum_{l, k} x_{l k} ⟨ g, ψ_{l k} ⟩$ ; Smooth functionals extend continuously to the white-noise topology ↗ proves that this series is absolutely convergent and defines a continuous linear functional on $H (δ)$ for every $δ > 0$ .

In the white-noise experiment $X^{(n)} = f_{0} + n^{- 1 / 2} W$ , fix $δ > 1 / 2$ so that the observation is $H (δ)$ -valued; $L_{g} (X^{(n)})$ means this continuous pairing with the $H (δ)$ -valued observation. It does not assert that $X^{(n)}$ is an $L^{2}$ function. The representer is nonzero when $g \neq 0$ ; the corresponding white-noise variance is $∥ g ∥_{2}^{2}$ .

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Type-S log-density and type-H histogram prior families

Definition

For type S use a fixed sufficiently regular compactly supported orthonormal multiresolution wavelet basis $(ψ_{l k})$ on $[0, 1]$ as in Logarithmic negative Sobolev spaces for white noise, with boundary correction, regularity greater than the truth smoothness $α$ , and $∣ K_{l} ∣ = 2^{l}$ . The spaces $V_{L}$ spanned by the retained levels $l \leq L$ contain constants and have orthonormal scaling bases with uniformly bounded overlap, support diameters at most $C 2^{- L}$ , and suprema at most $C 2^{L / 2}$ ; a fixed shift of the scaling-resolution index is absorbed in $C$ . The wavelets likewise have uniformly bounded overlap, support diameters at most $C 2^{- l}$ , and suprema at most $C 2^{l / 2}$ . For an integer $m > α$ , the detail wavelets have vanishing moments through degree $m - 1$ , and derivatives through order $m$ satisfy $∥ ψ_{l k}^{(q)} ∥_{\infty} \leq C_{q} 2^{l (q + 1 / 2)}$ , with uniformly bounded overlap of their supports; the fixed low scaling functions are $C^{m}$ . These are the sufficiently regular compact multiresolution basis conditions in the source density theorem. For the density-model source results, every truth $f_{0}$ is a probability density satisfying $0 < ρ_{0} \leq f_{0} \leq D_{0} < \infty$ for constants $ρ_{0}, D_{0}$ . For $n \geq 3$ and smoothness index $α > 0$ , let $j_{n}$ and $ℓ_{n}$ be the largest integers satisfying $2^{j_{n}} \leq n^{1 / (2 α + 1)}, 2^{ℓ_{n}} \leq (n / \log n)^{1 / (2 α + 1)} .$ Choose one cutoff sequence $L_{n} \in {j_{n}, ℓ_{n}}$ for all $n \geq 3$ ; set any finite $L_{n}$ for $n = 1, 2$ , which does not change asymptotic claims.

For a finite wavelet series $T_{n} = \sum_{l \leq L_{n}} \sum_{k \in K_{l}} σ_{l} ξ_{l k} ψ_{l k}$ , define $c (T_{n}) = \log \int_{0}^{1} e^{T_{n} (x)} d x$ and the normalized log-density $f_{n} (x) = e^{T_{n} (x) - c (T_{n})}$ .

The type-S1 family takes IID coefficients $ξ_{l k}$ with a continuous density $φ_{H}$ for which $\log φ_{H}$ is Lipschitz, and uses $σ_{l} = 2^{- l (α + 1 / 2)}$ . Its source theorem assumes $\log f_{0}$ has coefficient Hölder regularity $α > 1$ and $f_{0}$ is bounded above and bounded away from zero.

The type-S2 family takes IID standard-normal coefficients and uses $σ_{l} = 2^{- l (r + 1 / 2)}$ , where $α > 1 / 2$ and $0 < r < α - 1 / 4$ . Here the truth regime is $\log f_{0} \in C^{α}$ with positive finite density bounds. Equation (24) in the source specifies this prior-parameter range, but its Theorem 3 assumes $α > 1$ for both type-S families. The broader Gaussian truth range $α > 1 / 2$ is established separately by the local type-S posterior BvM theorem.

For the type-H family, use the Haar basis instead of the regular type-S wavelets: its scaling function is $1$ and $ψ_{l k} = 2^{l / 2} (1_{I_{l + 1, 2 k}} - 1_{I_{l + 1, 2 k + 1}})$ for $l \geq 0$ and $0 \leq k < 2^{l}$ . The histogram space at resolution $L$ consists of the scaling coordinate and detail levels $l < L$ ; all detail coefficients at $l \geq L$ vanish. Partition $[0, 1]$ into dyadic cells $I_{L_{n}, k}$ of length $2^{- L_{n}}$ , draw $ω = (ω_{k})_{k < 2^{L_{n}}} \sim Dirichlet (a_{0, L_{n}}, \dots, a_{2^{L_{n}} - 1, L_{n}})$ , and set $f_{n} = 2^{L_{n}} ω_{k}$ on cell $I_{L_{n}, k}$ . The source assumptions are $c_{1} 2^{- L_{n} a} \leq a_{k, L_{n}} \leq c_{2}$ for fixed positive $a, c_{1}, c_{2}$ , and $f_{0} \in C^{α}$ for $1 / 2 < α \leq 1$ , together with the common positive finite density bounds above. The posterior asymptotic conclusions for these families are separate results.

Here $C_{ψ}^{α}$ denotes the coefficient class with norm $∥ g ∥_{C_{ψ}^{α}} : = \sup_{l, k} 2^{l (α + 1 / 2)} ∣ ⟨ g, ψ_{l k} ⟩ ∣$ . For sufficiently regular standard wavelets this is the usual Hölder scale (with the usual Zygmund interpretation at integer smoothness).

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

The multiscale coefficient space is separable and has vanishing projection tails

Statement

For every weight sequence in Little-o multiscale coefficient space M0(w), $M_{0} (w)$ is a separable Banach space. If $P_{J}$ retains exactly the coefficients at levels $l \leq J$ , then $∥ P_{J} x ∥_{M (w)} \leq ∥ x ∥_{M (w)}$ and $∥ x - P_{J} x ∥_{M (w)} \to 0$ for each $x \in M_{0} (w)$ . Every square-summable coefficient sequence belongs to $M_{0} (w)$ .

Facts & Assumptions

Given: The finite level sets $K_{l}$ , weights $w_{l} \geq 1$ , and the vanishing-tail definition of $M_{0} (w)$ .

[F1]

Little-o multiscale coefficient space M0(w): the detail norm is $D (x) = \sup_{l} w_{l}^{- 1} \max_{k \in K_{l}} ∣ x_{l k} ∣$ . Without an extra scaling coordinate the norm is $D (x)$ ; in the Haar convention, writing $x_{*}$ for that coordinate, it is $\max (∣ x_{*} ∣, D (x))$ . Membership in $M_{0} (w)$ means the normalized detail maximum tends to zero. As in that definition, every $P_{J}$ retains the finite coarsest coordinates, including $x_{*}$ when present.

Proof

1.1F1givenchoosealgebra

Let $(x^{(m)})$ be Cauchy in $M_{0} (w)$ . Each coordinate is Cauchy because $∣ x_{l k}^{(m)} - x_{l k}^{(n)} ∣ \leq w_{l} ∥ x^{(m)} - x^{(n)} ∥_{M (w)}$ . In the Haar case also $∣ x_{*}^{(m)} - x_{*}^{(n)} ∣ \leq ∥ x^{(m)} - x^{(n)} ∥_{M (w)}$ . Let $x_{l k}$ and, when present, $x_{*}$ be the scalar limits. The uniform Cauchy bound passes to every finite collection of coordinates and then to the supremum, so $x^{(m)} \to x$ in the weighted supremum norm. To see $x \in M_{0} (w)$ , fix $ε > 0$ , choose $m$ with $∥ x - x^{(m)} ∥_{M (w)} < ε$ , then choose $J$ so the normalized tail of $x^{(m)}$ is below $ε$ ; for $l > J$ the normalized tail of $x$ is at most $2 ε$ . Thus $M_{0} (w)$ is complete.

1.2F1givenchoosealgebra

Finite-support sequences with rational coordinates are countable and dense: first choose $J$ to make the tail of any $x \in M_{0} (w)$ small, then approximate its finitely many coordinates through level $J$ , including the scaling coordinate when present, by rationals. Hence $M_{0} (w)$ is separable. The formula for the norm shows $P_{J}$ is a contraction, and the defining tail limit gives $∥ x - P_{J} x ∥_{M (w)} \to 0$ .

2.1F1givenalgebra∎

If $x \in ℓ^{2}$ , then $w_{l} \geq 1$ implies $w_{l}^{- 1} \max_{k} ∣ x_{l k} ∣ \leq (\sum_{k} ∣ x_{l k} ∣^{2})^{1 / 2}$ . The right side tends to zero because its squares are level blocks of a convergent nonnegative series. Its supremum and the absolute scaling coordinate, when present, are at most $∥ x ∥_{ℓ^{2}}$ , so $x \in M_{0} (w)$ .

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The multiscale primitive map is continuous into C[0,1]

Statement

Let $(ψ_{l k})$ be a compactly supported wavelet basis of Logarithmic negative Sobolev spaces for white noise. The Haar basis with its finite scaling coordinate, as specified in Little-o multiscale coefficient space M0(w), is also allowed. Assume each level- $l$ wavelet for $l \geq l_{0}$ has zero integral, uniformly bounded overlap, support length at most $C_{0} 2^{- l}$ , and $∥ ψ_{l k} ∥_{1} \leq C_{0} 2^{- l / 2}$ . Let $M_{0} (w)$ be as in Little-o multiscale coefficient space M0(w), and suppose $S_{w} : = \sum_{l \geq l_{0}} w_{l} 2^{- l / 2} < \infty .$ For $x \in M_{0} (w)$ define $J x (t) : = \sum_{l, k} x_{l k} ⟨ 1_{[0, t]}, ψ_{l k} ⟩_{L^{2}}, 0 \leq t \leq 1.$ Then this series converges absolutely and uniformly in $t$ , $J x \in C ([0, 1])$ , and $∥ J x - J y ∥_{\infty} \leq C S_{w} ∥ x - y ∥_{M (w)}$ after adding the finite coarsest-level constant to $C S_{w}$ . If $x$ is the coefficient sequence of $f \in L^{2}$ , then $J x (t) = \int_{0}^{t} f (u) d u$ .

Facts & Assumptions

Given: The wavelet geometry, summability condition, and multiscale space in the Statement.

[F1]

Little-o multiscale coefficient space M0(w): $∥ x ∥_{M (w)} = \sup_{l} w_{l}^{- 1} \max_{k} ∣ x_{l k} ∣$ .

[F2]

Logarithmic negative Sobolev spaces for white noise: $(ψ_{l k})$ is a complete orthonormal basis of $L^{2} ([0, 1])$ ; finite orthogonal projections converge in $L^{2}$ .

Proof

1.1givenalgebra

For fixed $t$ and level $l \geq l_{0}$ , a wavelet whose support lies wholly on one side of $t$ contributes zero: it is either disjoint from $[0, t]$ or its entire support is integrated and its mean is zero. Only supports meeting $t$ can contribute. Uniformly bounded overlap bounds their number by a constant, and each coefficient has magnitude at most $∥ ψ_{l k} ∥_{1} \leq C_{0} 2^{- l / 2}$ . Thus $\sup_{t} \sum_{k} ∣ ⟨ 1_{[0, t]}, ψ_{l k} ⟩ ∣ \leq C_{1} 2^{- l / 2}$ . The finitely many coarsest levels have a finite uniform sum as well.

2.1F1givenstep 1.1algebra

By [F1] and step 1.1, the absolute sum of all terms at levels $l \geq l_{0}$ is at most $C_{1} ∥ x ∥_{M (w)} \sum_{l \geq l_{0}} w_{l} 2^{- l / 2}$ , uniformly over $t$ . Its tails tend uniformly to zero because $S_{w} < \infty$ . Each finite-level sum is continuous in $t$ , since $t \mapsto \int_{0}^{t} ψ_{l k} (u) d u$ is continuous for an $L^{1}$ function. The uniform limit is continuous. Applying the same estimate to $x - y$ proves the stated operator-norm bound.

3.1F2step 2.1algebra∎

If $x_{l k} = ⟨ f, ψ_{l k} ⟩$ for $f \in L^{2}$ , the finite orthogonal projections converge to $f$ in $L^{2}$ by [F2]. The map $f \mapsto \int_{0}^{t} f (u) d u = ⟨ f, 1_{[0, t]} ⟩$ is a continuous $L^{2}$ functional because $∥ 1_{[0, t]} ∥_{2} \leq 1$ . Applying it to the finite projections shows that their coefficient sums converge to $\int_{0}^{t} f$ , while step 2.1 identifies the same limit with $J x (t)$ .

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Posterior quantiles stabilize under scalar weak convergence

Statement

Let $Q_{n} (y, \cdot)$ be data-indexed probability kernels on $R$ , let $Q$ be a fixed law with continuous CDF $F$ , and suppose $β_{B L} (Q_{n} (y, \cdot), Q) \to 0$ in probability. Fix $0 < u < 1$ and let $q = q_{u} (Q) : = \inf {r \in Q : Q ((- \infty, r]) \geq u}$ , using the canonical rational embedding. Assume $F$ is strictly increasing on some open interval containing $q$ . Define $q_{u} (Q_{n} (y, \cdot))$ by the same convention. Then $q_{u} (Q_{n}) \to q$ in probability. The local strict-increase hypothesis permits CDFs, such as norms of nonnegative random elements, that are constant on part of $R$ .

Facts & Assumptions

Given: The data-indexed kernels, fixed $Q$ , and $u \in (0, 1)$ in the Statement; $β_{B L} (Q_{n}, Q) = \sup {∣ \int h d Q_{n} - \int h d Q ∣ : ∥ h ∥_{\infty} \leq 1, Lip (h) \leq 1}$ .

[F1]

Posterior quantiles are measurable and bracket their probability level: the lower quantile is the real infimum of ${r \in Q : Q ((- \infty, r]) \geq u}$ and its CDF satisfies $Q ((- \infty, q)) \leq u \leq Q ((- \infty, q])$ .

[F2]

Continuity from above when one set has finite measure: probabilities of decreasing measurable sets with finite first measure converge to the probability of their intersection.

[A1]

The limiting CDF is continuous on $R$ and strictly increasing on an open interval containing $q_{u} (Q)$ .

Proof

1.1A1F2givenalgebra

Fix $x \in R$ and $η > 0$ . Choose $0 < δ < \min (1, η)$ . The piecewise-linear ramps that fall from $1$ to $0$ on $[x - δ, x]$ and on $[x, x + δ]$ bound $1_{(- \infty, x]}$ below and above. Multiplying each ramp by $δ$ makes its sup norm and Lipschitz constant at most $1$ . Comparing the lower and upper ramps under $Q_{n}$ with their integrals under $Q$ gives $∣ Q_{n} ((- \infty, x]) - F (x) ∣ \leq δ^{- 1} β_{B L} (Q_{n}, Q) + Q ([x - δ, x + δ])$ . By [A1], $Q ({x}) = 0$ ; [F2] then gives $Q ([x - δ, x + δ]) \to 0$ as $δ ↓ 0$ . First choose $δ$ small and then let $n \to \infty$ ; this proves $Q_{n} ((- \infty, x]) \to F (x)$ in probability.

2.1A1F1step 1.1algebra∎

Write $q = q_{u} (Q)$ as in [F1]. Since [A1] makes $F$ continuous, the CDF bracket in [F1] gives $F (q) = u$ . For every sufficiently small $ε > 0$ , strict increase near $q$ then gives $F (q - ε) < u < F (q + ε)$ . Step 1.1 at these two points implies, with probability tending to one, $Q_{n} ((- \infty, q - ε]) < u < Q_{n} ((- \infty, q + ε])$ . By monotonicity, no rational at or below $q - ε$ is in the defining set for $q_{u} (Q_{n})$ , while every rational above $q + ε$ belongs to it. Its infimum therefore lies in $[q - ε, q + ε]$ . Since arbitrarily small such $ε$ are available, the quantiles converge in probability.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Smooth functionals extend continuously to the white-noise topology

Statement

Let $H (δ)$ be the coefficient Hilbert space of Logarithmic negative Sobolev spaces for white noise, with weights $w_{l} = a_{l}^{- 1} (\log a_{l})^{- 2 δ}$ . If $g \in H_{2}^{s}$ for some $s > 1 / 2$ , then the pairing $L_{g} (x) = \sum_{l, k} x_{l k} g_{l k}$ , where $g_{l k} = ⟨ g, ψ_{l k} ⟩$ , converges absolutely for every $x \in H (δ)$ and is continuous, with $∣ L_{g} (x) ∣ \leq C_{s, δ} ∥ g ∥_{H_{2}^{s}} ∥ x ∥_{H (δ)} .$ On $L^{2} ([0, 1])$ this extension agrees with the usual inner product $⟨ f, g ⟩_{L^{2}}$ .

Facts & Assumptions

Given: The wavelet coefficient weights and the Sobolev coefficient norm in the Statement.

[F1]

Smooth linear functionals on the white-noise topology: $g \in H_{2}^{s}$ means $\sum_{l, k} a_{l}^{2 s} ∣ g_{l k} ∣^{2} < \infty$ , and its pairing with an $L^{2}$ function is the $L^{2}$ inner product.

[F2]

Logarithmic negative Sobolev spaces for white noise: $∥ x ∥_{H (δ)}^{2} = \sum_{l, k} w_{l} ∣ x_{l k} ∣^{2}$ , where $w_{l} = a_{l}^{- 1} (\log a_{l})^{- 2 δ}$ and the coarsest level has finite multiplicity.

Proof

1.1F1F2givenalgebra

Since $s > 1 / 2$ , the level factor $a_{l}^{1 - 2 s} (\log a_{l})^{2 δ}$ is bounded: it tends to zero as $a_{l} \to \infty$ , because a positive power of $a_{l}$ dominates every fixed power of $\log a_{l}$ , and the finitely many coarsest levels have finite weights. Thus $w_{l}^{- 1} \leq C_{s, δ}^{2} a_{l}^{2 s}$ after increasing a finite constant.

2.1F1F2step 1.1algebra

Weighted Cauchy–Schwarz gives $\sum_{l, k} ∣ x_{l k} g_{l k} ∣ \leq (\sum_{l, k} w_{l} ∣ x_{l k} ∣^{2})^{1 / 2} (\sum_{l, k} w_{l}^{- 1} ∣ g_{l k} ∣^{2})^{1 / 2} \leq C_{s, δ} ∥ x ∥_{H (δ)} ∥ g ∥_{H_{2}^{s}}$ . The same bound applied to $x - y$ proves continuity and absolute convergence of the series.

3.1F1F2step 2.1algebra∎

If $x$ is the coefficient sequence of $f \in L^{2}$ , its finite orthonormal projections converge to $f$ in $L^{2}$ because $(ψ_{l k})$ is a complete orthonormal basis. Their inner products with $g$ are the finite partial sums of the displayed pairing, and converge to $⟨ f, g ⟩_{L^{2}}$ by continuity of the inner product. Step 2.1 identifies the same limit with $L_{g} (x)$ .

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Finite type-S and type-H prior draws are L2 probability densities

Statement

Every finite type-S draw and every type-H draw in Type-S log-density and type-H histogram prior families is a well-defined nonnegative $L^{2} ([0, 1])$ probability density.

Facts & Assumptions

Given: A finite type-S coefficient array or a type-H Dirichlet vector, and the bounded finite-level wavelet basis and normalized prior constructions of Type-S log-density and type-H histogram prior families.

[F1]

Type-S log-density and type-H histogram prior families: type S sets $f = e^{T - c (T)}$ for a finite bounded wavelet sum $T$ , with $c (T) = \log \int_{0}^{1} e^{T}$ ; type H sets $f = 2^{L} ω_{k}$ on dyadic cells of length $2^{- L}$ , where $ω_{k} \geq 0$ and $\sum_{k} ω_{k} = 1$ .

Proof

1.1F1givenalgebra

For a type-S draw, boundedness of the finite wavelet sum gives $M = ∥ T ∥_{\infty} < \infty$ , hence $e^{- M} \leq e^{T (x)} \leq e^{M}$ on $[0, 1]$ . Therefore $0 < e^{- M} \leq \int_{0}^{1} e^{T (x)} d x \leq e^{M} < \infty$ , so $c (T)$ is finite and $f = e^{T - c (T)}$ is positive and integrates to one. The same bounds make $f$ bounded, so $f \in L^{2} ([0, 1])$ .

2.1F1givenalgebra∎

For type H, each dyadic cell has length $2^{- L}$ , so $\int_{0}^{1} f (x) d x = \sum_{k < 2^{L}} 2^{L} ω_{k} 2^{- L} = \sum_{k} ω_{k} = 1$ . Also $∥ f ∥_{2}^{2} = 2^{L} \sum_{k} ω_{k}^{2} \leq 2^{L} (\sum_{k} ω_{k})^{2} = 2^{L} < \infty$ , since all weights are nonnegative. Thus every type-H draw is a nonnegative $L^{2}$ probability density.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Finite-projection convergence and posterior tail control imply weak BvM

Statement

Let $Q_{n}$ be data-indexed probability measures on $M_{0} (w)$ and let $G$ be a fixed probability measure on this space. For the projections $P_{J}$ of The multiscale coefficient space is separable and has vanishing projection tails, suppose that for every fixed $J$ , $β_{B L} (Q_{n} \circ P_{J}^{- 1}, G \circ P_{J}^{- 1}) \to 0$ in probability, and that for every $ε, η > 0$ , $\lim_{J \to \infty} {lim sup}_{n \to \infty} P {Q_{n} (∥ x - P_{J} x ∥_{M (w)} > ε) > η} = 0.$ Then $β_{B L} (Q_{n}, G) \to 0$ in probability.

Facts & Assumptions

Given: The $M_{0} (w)$ norm and projections, the bounded-Lipschitz metric, and the finite-projection and posterior-tail assumptions in the Statement.

[F1]

Little-o multiscale coefficient space M0(w): $M_{0} (w)$ has the weighted supremum norm and consists of coefficient sequences whose normalized level maxima vanish.

[F2]

The multiscale coefficient space is separable and has vanishing projection tails: $P_{J} x \to x$ in $M_{0} (w)$ for every $x \in M_{0} (w)$ .

[F3]

Continuity from above when one set has finite measure: if measurable sets decrease and the first has finite measure, their measures decrease to the measure of their intersection.

Proof

1.1F1givenalgebra

For any probability measure $Q$ on $M_{0} (w)$ and any bounded $1$ -Lipschitz function $h$ , $∣ h (x) - h (P_{J} x) ∣ \leq \min (2, ∥ x - P_{J} x ∥_{M (w)})$ . Splitting at $ε > 0$ and taking the supremum over such $h$ gives $β_{B L} (Q, Q \circ P_{J}^{- 1}) \leq ε + 2 Q (∥ x - P_{J} x ∥_{M (w)} > ε)$ . Applying this to $Q_{n}$ and $G$ , then using the triangle inequality, bounds $β_{B L} (Q_{n}, G)$ by the two projection errors and the fixed- $J$ distance between projected laws.

2.1F2F3givenstep 1.1algebra∎

For each $x \in M_{0} (w)$ , [F2] gives $∥ x - P_{J} x ∥_{M (w)} \to 0$ . Thus the events ${x : ∥ x - P_{J} x ∥_{M (w)} > ε}$ decrease to the empty set, and [F3] makes their $G$ -probabilities tend to zero. Given any target error, first choose $ε$ small, then choose $J$ large so this $G$ -tail and the limiting posterior-tail probability are small; for that fixed $J$ , the projected-law distance tends to zero in probability by assumption. Step 1.1 then implies $β_{B L} (Q_{n}, G) \to 0$ in probability.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Linear-functional BvM limit and credible-interval calibration

Statement

Use the product-prior white-noise model and supplied posterior kernel of Weak nonparametric Bernstein–von Mises theorem for product priors, including its wavelet-basis, coefficient-density, truth, and countable-choice hypotheses. Let $g \in H_{2}^{s}$ for some $s > 1 / 2$ , with $g \neq 0$ , and put $σ_{g} = ∥ g ∥_{2}$ . Let $Q_{n}$ be the posterior law of $L_{g} (f)$ and let $C_{n, α} = [q_{α / 2} (Q_{n}), q_{1 - α / 2} (Q_{n})], 0 < α < 1.$ Then $β_{B L, R} ⁣ (Q_{n} \circ [u \mapsto \sqrt{n} (u - L_{g} (X^{(n)}))]^{- 1}, N (0, σ_{g}^{2})) ⟶ 0$ in $P_{f_{0}}^{(n)}$ -probability. The interval $C_{n, α}$ is posterior credible at level $1 - α$ , and its frequentist coverage satisfies $P_{f_{0}}^{(n)} {L_{g} (f_{0}) \in C_{n, α}} ⟶ 1 - α .$

Facts & Assumptions

Given: The white-noise experiment, product prior, posterior kernel, basis, density and truth assumptions of Weak nonparametric Bernstein–von Mises theorem for product priors, together with $g \neq 0$ , $g \in H_{2}^{s}$ , $s > 1 / 2$ , and $0 < α < 1$ .

[F1]

Weak nonparametric Bernstein–von Mises theorem for product priors: for every $δ > 1 / 2$ , the posterior law of $\sqrt{n} (f - X^{(n)})$ converges in bounded-Lipschitz distance in $H (δ)$ to the canonical white-noise law $N$ in $P_{f_{0}}^{(n)}$ -probability. Its product-prior construction requires countable choice; the same axiom suffices for the scalar density-normal interface.

[F2]

Smooth functionals extend continuously to the white-noise topology: $L_{g}$ is a continuous linear functional on $H (δ)$ and agrees with $f \mapsto ⟨ f, g ⟩_{L^{2}}$ on $L^{2}$ .

[F3]

White noise is a Borel Gaussian law on the negative Sobolev space: the canonical law $N$ is centered Gaussian on $H (δ)$ ; for the coefficient pairing $L_{g}$ , its variance is $\sum_{l, k} ∣ g_{l k} ∣^{2} = ∥ g ∥_{2}^{2}$ .

[F4]

Posterior quantiles stabilize under scalar weak convergence: bounded-Lipschitz convergence in probability to a law with continuous CDF locally strictly increasing at an interior quantile implies convergence in probability of that posterior quantile.

[F5]

Equal-tailed posterior quantile intervals are credible: the equal-tailed interval between posterior quantiles $q_{α / 2}$ and $q_{1 - α / 2}$ is graph-measurable and has posterior mass at least $1 - α$ .

[F6]

Density normal laws and square-summable Gaussian series under countable choice: Under countable choice, in its density convention, $N (0, 1)$ is defined by its continuous positive density, and $N (0, σ^{2})$ for $σ > 0$ is its image under $x \mapsto σ x$ ; change of variables gives a continuous strictly positive density and hence a continuous strictly increasing CDF.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice is inherited through [F1] for the $L^{2}$ product-prior construction and also suffices for the density normal-law interface [F6]; no full choice is used below.

Proof

1.1A1F1F2F3algebra

Fix $δ > 1 / 2$ . For every bounded $1$ -Lipschitz function $h : R \to R$ , the composition $h \circ L_{g}$ is bounded by $1$ and has Lipschitz constant at most $∥ L_{g} ∥_{H (δ)^{*}}$ . Rescaling by $c_{g} = \max (1, ∥ L_{g} ∥_{H (δ)^{*}})$ makes it an admissible bounded-Lipschitz test on $H (δ)$ . Hence the scalar pushforward distance is at most $c_{g}$ times the distance in [F1]. The pushforward of $N$ is $N (0, σ_{g}^{2})$ by [F3], proving the asserted scalar BvM limit.

2.1F4F5F6step 1.1algebra

Translate the posterior law of $L_{g} (f)$ by $L_{g} (X^{(n)})$ and scale by $\sqrt{n}$ . By [F4] and the continuous strictly increasing normal CDF from [F6], the two quantiles of this scaled law converge in probability to $σ_{g} z_{α / 2}$ and $σ_{g} z_{1 - α / 2}$ , where $z_{u}$ is the $u$ -quantile of $N (0, 1)$ . Since the affine map is increasing, these are exactly the scaled endpoints of $C_{n, α} - L_{g} (X^{(n)})$ . The interval is credible by [F5].

3.1F3F6step 2.1algebra

Under the true white-noise law, $\sqrt{n} {L_{g} (X^{(n)}) - L_{g} (f_{0})} = L_{g} (W)$ has exactly the centered normal law $N (0, σ_{g}^{2})$ by [F3]. On the event that both random scaled endpoints in step 2.1 lie within $ε$ of their limits, the event $L_{g} (f_{0}) \in C_{n, α}$ is bounded between the corresponding deterministic normal-interval events with endpoints enlarged or contracted by $ε$ . This sandwich uses no independence between the endpoints and the observation. The endpoint event has probability tending to one, and [F6] makes the normal mass of shrinking neighborhoods of the two endpoints tend to zero.

4.1F6step 3.1algebra∎

Letting $n \to \infty$ and then $ε ↓ 0$ in the sandwich of step 3.1 gives limiting coverage $Φ (z_{1 - α / 2}) - Φ (z_{α / 2}) = 1 - α$ by the definition of the normal quantiles and CDF in [F6].

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Nonlinear-functional BvM limit under fast L2 contraction

Statement

Use the white-noise experiment, product prior, and supplied posterior kernel of Weak nonparametric Bernstein–von Mises theorem for product priors, with all its stated basis, coefficient-density, truth, and countable-choice hypotheses. Let $Ψ : L^{2} ([0, 1]) \to R$ be Borel measurable. Suppose $f_{0} \in L^{2}$ , and for some $g \in H_{2}^{s}$ , $s > 1 / 2$ , $g \neq 0$ , $C < \infty$ , and $ρ > 0$ , $∣ Ψ (f_{0} + h) - Ψ (f_{0}) - ⟨ h, g ⟩_{L^{2}} ∣ \leq C ∥ h ∥_{2}^{2} whenever ∥ h ∥_{2} \leq ρ .$ Assume the posterior contracts in $L^{2}$ at a deterministic radius $r_{n} = o (n^{- 1 / 2})$ , meaning $Π_{n} (∥ f - f_{0} ∥_{2} > r_{n} ∣ X^{(n)}) = o_{P_{f_{0}}^{(n)}} (1)$ . Define the efficient center $θ_{n}^{*} : = Ψ (f_{0}) + L_{g} (X^{(n)} - f_{0}),$ where the last pairing is the continuous $H (δ)$ extension from Smooth functionals extend continuously to the white-noise topology. Then the posterior law of $\sqrt{n} {Ψ (f) - θ_{n}^{*}}$ converges in bounded-Lipschitz distance to $N (0, ∥ g ∥_{2}^{2})$ in $P_{f_{0}}^{(n)}$ -probability. The equal-tailed posterior interval for $Ψ (f)$ at level $1 - α$ is credible and has frequentist coverage tending to $1 - α$ .

The stipulated radius implies the source's squared-radius condition: with $s_{n} = r_{n}^{2}$ , one has $s_{n} = o (n^{- 1 / 2})$ and posterior mass inside $∥ f - f_{0} ∥_{2}^{2} \leq s_{n}$ tends to one. However, for $g \neq 0$ this radius is incompatible with the imported weak BvM limit. Thus the implication under the retained radius is vacuous; the source's squared-radius condition does not have this incompatibility.

Facts & Assumptions

Given: The product-prior weak BvM model of [F1], the Borel functional $Ψ$ , the stated local quadratic expansion with nonzero Riesz representer $g$ , the posterior $L^{2}$ radius $r_{n} = o (n^{- 1 / 2})$ , and $0 < α < 1$ .

[F1]

Weak nonparametric Bernstein–von Mises theorem for product priors: the posterior law of $\sqrt{n} (f - X^{(n)})$ converges in bounded-Lipschitz distance on $H (δ)$ to the canonical white-noise law for every $δ > 1 / 2$ .

[F2]

Smooth functionals extend continuously to the white-noise topology: $L_{g}$ extends continuously from $L^{2}$ to $H (δ)$ .

[F3]

White noise is a Borel Gaussian law on the negative Sobolev space: $L_{g} (W)$ is exactly $N (0, ∥ g ∥_{2}^{2})$ under the true white-noise law.

[F4]

Posterior quantiles stabilize under scalar weak convergence: scalar bounded-Lipschitz convergence to a CDF continuous on $R$ and locally strictly increasing at the target quantile implies quantile convergence.

[F5]

Equal-tailed posterior quantile intervals are credible: equal-tailed posterior quantiles define a measurable credible interval of posterior mass at least $1 - α$ .

[F6]

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice supplies the $L^{2}$ prior construction in [F1] and the density normal-law interface [F6]; no full choice is used below.

Proof

1.1givenalgebra

On the posterior event $∥ f - f_{0} ∥_{2} \leq r_{n} \leq ρ$ , write $R (f) = Ψ (f) - Ψ (f_{0}) - L_{g} (f - f_{0})$ . The expansion gives $\sqrt{n} ∣ R (f) ∣ \leq C \sqrt{n} r_{n}^{2} \to 0$ . Outside the event, posterior mass is $o_{P_{f_{0}}^{(n)}} (1)$ by assumption. Hence for every $ε > 0$ , $Π_{n} (\sqrt{n} ∣ R (f) ∣ > ε ∣ X^{(n)}) \to 0$ in $P_{f_{0}}^{(n)}$ -probability.

2.1A1F1F2F3step 1.1algebra

For posterior draws, $\sqrt{n} {Ψ (f) - θ_{n}^{*}} = \sqrt{n} L_{g} (f - X^{(n)}) + \sqrt{n} R (f)$ . For $c_{g} = \max (1, ∥ L_{g} ∥)$ , composing a bounded $1$ -Lipschitz scalar test with $L_{g}$ and dividing by $c_{g}$ gives an admissible bounded-Lipschitz test on $H (δ)$ . Thus the first term has bounded-Lipschitz distance to the pushforward limit at most $c_{g}$ times the distance in [F1]; by [F3] that limit is $N (0, ∥ g ∥_{2}^{2})$ . For any bounded $1$ -Lipschitz test function, replacing the first term by its sum with the remainder changes its posterior integral by at most $ε + 2 Π_{n} (\sqrt{n} ∣ R (f) ∣ > ε ∣ X^{(n)})$ . Step 1.1 and then $ε ↓ 0$ prove the asserted scalar BvM limit.

3.1F4F5F6step 2.1algebra

Apply [F4] to the centered and scaled posterior law in step 2.1. By [F6], its $α / 2$ and $1 - α / 2$ quantiles converge in probability to $∥ g ∥_{2} z_{α / 2}$ and $∥ g ∥_{2} z_{1 - α / 2}$ . Translating and rescaling these quantiles gives the equal-tailed posterior interval for $Ψ (f)$ ; its credibility follows from [F5].

4.1F3F6step 3.1algebra

Under the true experiment, $\sqrt{n} {θ_{n}^{*} - Ψ (f_{0})} = L_{g} (W)$ has exactly the law $N (0, ∥ g ∥_{2}^{2})$ by [F3]. The random centered interval endpoints in step 3.1 converge in probability to the corresponding normal quantiles. For any fixed $ε > 0$ , on the event that both endpoints are within $ε$ of their limits, coverage is bounded by deterministic normal-interval probabilities with endpoints enlarged or contracted by $ε$ . This argument requires no independence between the endpoints and $L_{g} (W)$ . By [F6], the mass of shrinking endpoint neighborhoods vanishes; letting $n \to \infty$ and then $ε ↓ 0$ gives coverage $1 - α$ .

5.1F1F2F3F6step 2.1givenconstructalgebra∎

To verify the caveat, put $Q = N (0, ∥ g ∥_{2}^{2})$ , whose density is bounded by $M = (2 π ∥ g ∥_{2}^{2})^{- 1 / 2}$ by [F6]. Choose $0 < a \leq 1$ with $2 a M < 1 / 2$ and set $c_{n} = - \sqrt{n} L_{g} (X^{(n)} - f_{0})$ . For each realized data set, $h_{n} (x) = (a - ∣ x - c_{n} ∣)_{+}$ is a bounded $1$ -Lipschitz test. On $∥ f - f_{0} ∥_{2} \leq r_{n}$ , its value at $\sqrt{n} L_{g} (f - X^{(n)})$ is at least $a - \sqrt{n} ∥ g ∥_{2} r_{n}$ , so its posterior integral tends to $a$ in probability. But $\int h_{n} d Q \leq 2 a^{2} M < a / 2$ for every center $c_{n}$ . The bounded-Lipschitz distance is a supremum over all such tests for each data set, so it cannot tend to zero. This contradicts the linear pushforward of [F1] established in step 2.1 and proves the incompatibility.

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Projected empirical bridge CLT in the little-o multiscale space

Statement

Assume AC. Let $P_{0}$ have a density $f_{0}$ on $[0, 1]$ with $∥ f_{0} ∥_{\infty} \leq B < \infty$ . Let $(ψ_{l k})$ be an orthonormal, compactly supported wavelet basis with $2^{l}$ coefficients at level $l \geq 1$ , uniformly bounded overlap, and $∥ ψ_{l k} ∥_{\infty} \leq C_{0} 2^{l / 2}$ . Suppose for some $γ > 1 / 2$ , $\max_{k} ∣ ⟨ f_{0}, ψ_{l k} ⟩ ∣ \leq C_{f} 2^{- l (γ + 1 / 2)} .$ Let $w_{l} \geq 1$ be nondecreasing and admissible in Little-o multiscale coefficient space M0(w), and choose either cutoff regime $2^{L_{n}} ≍ n^{1 / (2 γ + 1)} or 2^{L_{n}} ≍ (n / \log n)^{1 / (2 γ + 1)} .$ For IID $X_{1}, \dots, X_{n} \sim P_{0}$ , define the finite wavelet projection $P_{n} (L_{n})$ by coefficients $⟨ P_{n} (L_{n}), ψ_{l k} ⟩ = P_{n} ψ_{l k} : = n^{- 1} \sum_{i = 1}^{n} ψ_{l k} (X_{i})$ for $l \leq L_{n}$ and zero for $l > L_{n}$ , with the coarsest scaling coordinates projected as well. Let $G_{P_{0}}$ be a Borel centered Gaussian law on $M_{0} (w)$ whose coefficient covariance is $Cov (G_{l k}, G_{l^{'} k^{'}}) = P_{0} [(ψ_{l k} - P_{0} ψ_{l k}) (ψ_{l^{'} k^{'}} - P_{0} ψ_{l^{'} k^{'}})] .$ Then $\sqrt{n} {P_{n} (L_{n}) - f_{0}} \Rightarrow G_{P_{0}} in M_{0} (w) .$

Facts & Assumptions

Given: AC, the density and wavelet bounds, the coefficient smoothness and cutoff conditions, IID observations, and the stated Borel Gaussian bridge law.

[F1]

Little-o multiscale coefficient space M0(w): $M_{0} (w)$ uses the norm $\sup_{l} w_{l}^{- 1} \max_{k} ∣ x_{l k} ∣$ , with normalized level maxima tending to zero.

[F2]

The multiscale coefficient space is separable and has vanishing projection tails: square-summable coefficient sequences lie in $M_{0} (w)$ , and finite-level projections converge in its norm.

[F3]

Finite-projection convergence and posterior tail control imply weak BvM: convergence of every fixed finite projection together with vanishing tails under the approximating and limit laws implies bounded-Lipschitz convergence on $M_{0} (w)$ .

[F4]

Multivariate iid central limit theorem: under AC, finite-second-moment IID vectors have the multivariate normal limit with their covariance, including singular covariance.

[F5]

The bounded-Lipschitz metric characterizes weak convergence on Polish spaces: on finite-dimensional Euclidean space, weak convergence of Borel probabilities implies convergence in bounded-Lipschitz distance.

[F6]

Finite and countable subadditivity of measures: a probability of a finite union is bounded by the sum of the component probabilities.

[F7]

Markov's inequality for random variables: for nonnegative $Y$ and $a > 0$ , $\Pr (Y \geq a) \leq E [Y] / a$ .

[F8]

Continuity from above when one set has finite measure: probabilities of decreasing measurable sets with finite first measure converge to the probability of their intersection.

[F9]

Identical distribution and IID families: IID means the observations are mutually independent and have a common law.

[A1]

The Axiom of Choice: AC is the stated axiom used by [F4] for finite-dimensional CLTs; no common null set over all directions is required because each projection level $J$ is fixed.

[A2]

The basis satisfies the uniform support, overlap, coefficient-count and sup-norm bounds in the Statement.

[A3]

The truth coefficients satisfy the displayed $C_{f} 2^{- l (γ + 1 / 2)}$ bound.

Proof

Given: The probability law, basis, coefficient smoothness, admissible weights, cutoff, Gaussian limit law, and AC hypothesis in the Statement.

1.1F2givenalgebra

$P_{n} (L_{n})$ has finitely many nonzero coefficients, so it is in $M_{0} (w)$ . The coefficient sequence of $f_{0}$ is square-summable by Parseval, so [F2] places it in $M_{0} (w)$ . Thus $U_{n} = \sqrt{n} {P_{n} (L_{n}) - f_{0}}$ is a well-defined $M_{0} (w)$ -valued random element.

1.2A1A2F4F5F9given

Fix $J$ . For all sufficiently large $n$ , $L_{n} \geq J$ , so the coordinates of $P_{J} U_{n}$ are $n^{- 1 / 2} \sum_{i = 1}^{n} (ψ_{l k} (X_{i}) - P_{0} ψ_{l k})$ for the finitely many indices through level $J$ . The vector summand is bounded by [A2], has finite second moment, and is IID by [F9]; its covariance is the displayed covariance of $G_{P_{0}}$ . By [F4] these finite-dimensional vectors converge weakly to the corresponding Gaussian projection. The projection space is finite-dimensional Euclidean with an equivalent weighted maximum norm, so [F5] gives convergence in bounded-Lipschitz distance for every fixed $J$ .

1.3A2F7givenalgebra

Let $Y_{i} = ψ_{l k} (X_{i}) - P_{0} ψ_{l k}$ . Then $E Y_{i} = 0$ , $E Y_{i}^{2} \leq P_{0} ψ_{l k}^{2} \leq B$ , and $∣ Y_{i} ∣ \leq 2 C_{0} 2^{l / 2} = : b_{l}$ by [A2]. For $m \geq 2$ , $E ∣ Y_{i} ∣^{m} \leq b_{l}^{m - 2} E Y_{i}^{2}$ . Since $m! \geq 2 3^{m - 2}$ , expansion of the exponential gives, for $0 < λ b_{l} < 3 \sqrt{n}$ , $E \exp (λ n^{- 1 / 2} \sum_{i = 1}^{n} Y_{i}) \leq \exp (λ^{2} B / [2 (1 - λ b_{l} / (3 \sqrt{n}))])$ . Applying [F7] to this exponential and optimizing at $λ = t / (B + b_{l} t / (3 \sqrt{n}))$ yields $P_{0}^{n} {∣ \sqrt{n} (P_{n} - P_{0}) ψ_{l k} ∣ > t} \leq 2 \exp (- t^{2} / [2 (B + b_{l} t / (3 \sqrt{n}))])$ . Applying the same argument to $- Y_{i}$ supplies the lower-tail bound.

1.4A3F1givenalgebra

Above the cutoff, $U_{n}$ has only the deterministic bias coefficients $- \sqrt{n} ⟨ f_{0}, ψ_{l k} ⟩$ . By [A3] and monotonicity of $w_{l}$ , $\sup_{l > L_{n}, k} \sqrt{n} ∣ ⟨ f_{0}, ψ_{l k} ⟩ ∣ / w_{l} \leq C_{f} \sqrt{n} 2^{- L_{n} (γ + 1 / 2)} / w_{L_{n}}$ . For the first cutoff regime the numerator is bounded, while $w_{L_{n}} \to \infty$ . For the second, the numerator is $O (\sqrt{\log n})$ and $\sqrt{L_{n}} ≍ \sqrt{\log n}$ ; admissibility $w_{L_{n}} / \sqrt{L_{n}} \to \infty$ again makes the ratio vanish.

2.1F1F6step 1.3givenalgebra

Fix $ε > 0$ . By [F6] and step 1.3, $P_{0}^{n} {\max_{J < l \leq L_{n}, k} ∣ \sqrt{n} (P_{n} - P_{0}) ψ_{l k} ∣ / w_{l} > ε} \leq \sum_{J < l \leq L_{n}} 2^{l + 1} \exp (- ε^{2} w_{l}^{2} / [2 (B + (2 C_{0} ε / 3) 2^{l / 2} w_{l} / \sqrt{n})])$ . For levels with $2^{l / 2} w_{l} / \sqrt{n} \leq 1$ , the denominator is bounded by a constant depending only on $B, C_{0}, ε$ . Their sum is bounded by $\sum_{l > J} 2^{l + 1} e^{- c_{ε} w_{l}^{2}}$ , which tends to zero as $J \to \infty$ : admissibility gives $w_{l}^{2} / l \to \infty$ , so the summand is eventually bounded by $2^{- l}$ .

3.1step 2.1givenalgebra

For the remaining levels $2^{l / 2} w_{l} / \sqrt{n} > 1$ , the exponent in step 2.1 is at least $c_{ε} w_{l} \sqrt{n} / 2^{l / 2} \geq c_{ε} \sqrt{n} / 2^{L_{n} / 2}$ . Their total contribution is at most $2^{L_{n} + 2} \exp (- c_{ε} \sqrt{n} / 2^{L_{n} / 2})$ , which tends to zero. Indeed $L_{n} = O (\log n)$ and either cutoff gives $\sqrt{n} / 2^{L_{n} / 2} \geq c n^{γ / (2 γ + 1)}$ for all large $n$ after decreasing $c > 0$ if needed; this polynomial growth dominates $L_{n} \log 2$ .

4.1step 2.1step 3.1algebra

The two threshold classes partition all levels $J < l \leq L_{n}$ . The low-threshold sum tends to zero as $J \to \infty$ by step 2.1, and the high-threshold sum tends to zero as $n \to \infty$ by step 3.1. Therefore the centered empirical-fluctuation tails vanish in the iterated limit $n \to \infty$ , then $J \to \infty$ .

5.1F2F8step 4.1step 1.4

For the fixed Gaussian law $G_{P_{0}}$ , [F2] gives $∥ G_{P_{0}} - P_{J} G_{P_{0}} ∥_{M (w)} \to 0$ pointwise. Hence the events ${∥ G_{P_{0}} - P_{J} G_{P_{0}} ∥_{M (w)} > ε}$ decrease to the empty set; [F8] makes their probabilities tend to zero. Steps 4.1 and 1.4 give the matching posterior-tail condition for the laws of $U_{n}$ .

6.1F3step 1.2step 4.1step 1.4step 5.1∎

For each fixed $J$ , step 1.2 gives convergence of the projected laws in bounded-Lipschitz distance; steps 4.1 and 1.4 give vanishing tails for $U_{n}$ , and step 5.1 gives vanishing tails for $G_{P_{0}}$ . Applying [F3] to these laws proves $β_{B L} (L (U_{n}), G_{P_{0}}) \to 0$ , which is the asserted weak convergence on $M_{0} (w)$ .

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A bounded-density Gaussian bridge has a Borel multiscale law

Statement

Assume AC. Let $P_{0}$ have density $0 \leq f_{0} \leq D < \infty$ on $[0, 1]$ , and let $(ψ_{l k})$ be an orthonormal wavelet system with at most $C 2^{l}$ coordinates per level, including finitely many coarsest scaling coordinates. The Haar system is allowed. For every admissible $w$ , there exists a Borel centered Gaussian random element $G_{P_{0}}$ of $M_{0} (w)$ with coefficient covariance $K ((l, k), (l^{'}, k^{'})) = P_{0} [(ψ_{l k} - P_{0} ψ_{l k}) (ψ_{l^{'} k^{'}} - P_{0} ψ_{l^{'} k^{'}})] .$ Its Borel law is uniquely determined by these finite-coordinate Gaussian laws.

Facts & Assumptions

Given: AC, the bounded density, orthonormal system, finite level multiplicities, and admissible weights in the Statement.

[F1]

A mean and positive-semidefinite kernel define a Gaussian process constructs a Gaussian coordinate process for any symmetric positive-semidefinite kernel under AC.

[F2]

The multiscale coefficient space is separable and has vanishing projection tails: $M_{0} (w)$ is a separable Banach space; its coordinate projections $P_{J}$ are contractions and converge pointwise in norm to the identity.

[F3]

Characteristic function of a normal law and Uniqueness of a law from its characteristic function: centered scalar normals have characteristic function $e^{- v t^{2} / 2}$ and are determined by it under AC.

[F4]

Markov's inequality for random variables and Finite and countable subadditivity of measures give exponential tail and union bounds.

[F5]

Dominated convergence passes bounded almost-sure limits through expectation.

Proof

Given: The hypotheses in the Statement, including full AC.

1.1F1givenalgebra

The displayed kernel is symmetric and positive semidefinite because for any finite coefficients $a_{j}$ , $\sum_{i, j} a_{i} a_{j} K (i, j) = P_{0} [\sum_{j} a_{j} (ψ_{j} - P_{0} ψ_{j})]^{2} \geq 0$ . Thus [F1] constructs a centered jointly Gaussian coordinate sequence $Z$ with this covariance. Each coordinate variance is at most $P_{0} ψ_{l k}^{2} \leq D$ .

2.1F3F4step 1.1algebra

For a centered normal of variance $v \leq D$ , completing the square in its normal density gives $E e^{t Z} = e^{v t^{2} / 2}$ ; at $v = 0$ the variable is zero almost surely. Exponential Markov with $t = u / D$ therefore gives $\Pr (∣ Z ∣ > u) \leq 2 e^{- u^{2} / (2 D)}$ when $D > 0$ , as it is for a probability density. Consequently $\Pr {\max_{k} ∣ Z_{l k} ∣ > R \sqrt{l + 1}} \leq 2 C 2^{l} e^{- R^{2} (l + 1) / (2 D)}$ . Choose $R$ with $R^{2} / (2 D) > 2 \log 2$ . The sum over $l$ is finite, and the union bound over $l \geq J$ tends to zero. The probability that infinitely many of these events occur is therefore zero, since that event is contained in every such tail union.

3.1F2step 2.1constructalgebra

Almost surely the normalized maxima $\max_{k} ∣ Z_{l k} ∣ / w_{l}$ tend to zero, by step 2.1 and admissibility $w_{l} / \sqrt{l + 1} \to \infty$ ; each of the finitely many remaining coordinates is finite. The event $E = {Z \in M_{0} (w)}$ is coordinate-measurable, since the little-o condition is expressed by countable unions and intersections of the inequalities $\max_{k} ∣ Z_{l k} ∣ / w_{l} \leq 1 / q$ . It has probability one. Define $G = Z$ on $E$ and $G = 0$ otherwise. Its finite projections are measurable finite-dimensional random elements and converge everywhere in the separable Banach space [F2]. Hence $G$ is Borel measurable: for every closed set $F$ , the measurable functions $d (P_{J} G, F)$ converge to $d (G, F)$ ; inverse images of closed sets are thus measurable, and closed sets generate the Borel sigma-algebra.

4.1F2F3F5step 3.1algebra

Let $L$ be any continuous real linear functional on $M_{0} (w)$ . Then $L (P_{J} G) \to L (G)$ almost surely, and each approximant is centered normal with some variance $v_{J}$ . These variances are bounded. Indeed, almost-sure convergence makes the laws tight: choose $M$ so that $\Pr {\sup_{J} ∣ L (P_{J} G) ∣ > M} < 1 / 2$ , using that the supremum is finite almost surely. If $v_{J} \to \infty$ on a subsequence, the normal density bound would give $\Pr (∣ L (P_{J} G) ∣ \leq M) \leq 2 M / \sqrt{2 π v_{J}} \to 0$ , a contradiction. Any convergent subsequence of $(v_{J})$ with limit $v$ gives, by [F3] and [F5], $E e^{i t L (G)} = e^{- v t^{2} / 2}$ . Taking $t = 1$ shows that all subsequential variance limits coincide. Thus $v_{J} \to v \geq 0$ , and [F3] identifies $L (G)$ as $N (0, v)$ . Therefore $G$ is a centered Borel Gaussian random element.

5.1F1F2step 3.1step 4.1algebra∎

Its coordinate covariance and finite Gaussian laws are unchanged by the null-set modification in step 3.1. Finally the Borel sigma-algebra of $M_{0} (w)$ equals the sigma-algebra generated by its countably many coordinates: the norm is their countable weighted supremum, so all open balls about a countable dense set are coordinate-measurable, and these balls generate the topology. Thus two Borel laws with the same finite-coordinate laws agree, by uniqueness of the cylinder law in [F1] restricted to this measurable space. This proves existence and uniqueness of the stated bridge law.

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Continuous and strictly increasing norm distributions for density bridges

Statement

Assume AC. Let $P_{0}$ have density $0 < ρ \leq f_{0} \leq D < \infty$ on $[0, 1]$ , and put $F_{0} (t) = \int_{0}^{t} f_{0}$ . Use a complete real orthonormal wavelet basis, allowing the Haar basis with its finite scaling coordinate, and admissible weights $w$ as in Little-o multiscale coefficient space M0(w). Assume the wavelet geometry and summability conditions of The multiscale primitive map is continuous into C[0,1], and write $J : M_{0} (w) \to C ([0, 1])$ for its primitive map. Let $G$ be the Borel centered Gaussian bridge random element of A bounded-density Gaussian bridge has a Borel multiscale law, with coefficient covariance $Cov (G_{l k}, G_{l^{'} k^{'}}) = P_{0} [(ψ_{l k} - P_{0} ψ_{l k}) (ψ_{l^{'} k^{'}} - P_{0} ψ_{l^{'} k^{'}})] .$ Then the distribution functions of $∥ G ∥_{M (w)}$ and $∥ J G ∥_{\infty}$ are continuous on $R$ and strictly increasing on $(0, \infty)$ . Both assign probability zero to $0$ , and take values strictly between $0$ and $1$ at every positive argument. Consequently, for every $u \in (0, 1)$ , each has a unique positive finite lower $u$ -quantile $q$ , with CDF value $u$ , and is strictly increasing on an open neighborhood of $q$ .

Facts & Assumptions

Given: AC, the bounded positive density, complete wavelet basis, admissible weights, primitive map and Gaussian bridge in the Statement.

[F1]

The multiscale coefficient space is separable and has vanishing projection tails and The multiscale primitive map is continuous into C[0,1]: $M_{0} (w)$ is separable, finite-level projections converge in its norm, and $J$ is continuous linear into $C ([0, 1])$ .

[F2]

Logarithmic negative Sobolev spaces for white noise: the basis is complete and orthonormal in $L^{2} ([0, 1])$ , so its finite orthogonal projections converge in $L^{2}$ .

[F3]

A bounded-density Gaussian bridge has a Borel multiscale law supplies the Borel centered Gaussian law and its finite coefficient covariances. A finite stack of independent Gaussian vectors is jointly Gaussian and Mean and covariance determine Gaussian finite-dimensional laws identify finite Gaussian laws, including singular laws, by means and covariances under AC.

[F4]

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique supplies two independent copies of each Borel probability. Finite measures agreeing on a generating pi-system and on the whole space are equal identifies Borel laws from their finite-coordinate laws whenever those coordinates generate the Borel sigma-algebra.

[F5]

Finite and countable subadditivity of measures and Continuity from above when one set has finite measure give the countable null-union rule and continuity of probabilities along decreasing measurable events.

[F6]

Cauchy-Schwarz for random variables controls the difference of second moments when both factors converge in $L^{2}$ .

[F7]

Under AC, standard Brownian motion $B$ exists by Existence of continuous Brownian motion, and put $β_{t} = B_{t} - t B_{1}$ . The A-page definitions Brownian motion and Gaussian process, finite expectation linearity in Linearity, monotonicity, and the modulus bound for expectation, and Covariance is symmetric and bilinear in finite linear combinations give the direct bridge verification in step 4.2. Use the everywhere-continuous zero-start representative of Brownian reflection principle.

[F8]

Uncorrelated jointly Gaussian blocks are independent: finite jointly Gaussian blocks with zero cross-covariance are independent, including singular blocks.

[F9]

Brownian reflection principle: for $a > 0$ and $z \leq a$ , $P (\max_{t \leq 1} B_{t} \geq a, B_{1} \leq z) = P (B_{1} \geq 2 a - z)$ .

[A1]

The Axiom of Choice: full AC is assumed for [F3], [F4], [F7]–[F9].

Proof

1.1A1F1F3F4givenconstruct

In either $E = M_{0} (w)$ or $E = C ([0, 1])$ , the Borel sigma-algebra is generated by countably many continuous linear coordinates: the wavelet coordinates in the first case, and evaluations at rational times in the second. Indeed, the norm distance to a fixed point is their countable supremum, with the weights included in the first case; in the second case continuity makes the rational supremum equal the full supremum. Separability then gives a countable base of coordinate-measurable balls. Separability of $M_{0} (w)$ is [F1]; rational polygonal functions give a countable dense family in $C ([0, 1])$ by uniform continuity and finite rational approximation. Thus finite-coordinate laws determine Borel laws by [F4]. If $X$ is any centered Borel Gaussian random element in either space and $X^{'}$ an independent copy, [F3] gives $- X \overset{d}{=} X$ and $X + X^{'} \overset{d}{=} \sqrt{2} X$ : every finite coordinate vector on either side is centered Gaussian with the same covariance.

2.1F4F5step 1.1constructalgebra

For the law $μ$ of such an $X$ , let $S$ be its support, the set of points whose every open neighborhood has positive probability. Its complement is the union of all zero-mass balls in a countable base, hence has measure zero by [F5]; $S$ is closed and nonempty. Symmetry from step 1.1 gives $S = - S$ . The support of $X + X^{'}$ is $\overline{S + S}$ : independent positive-mass balls around $s, t \in S$ give positive mass to every neighborhood of $s + t$ , proving one inclusion; the other follows because $X, X^{'} \in S$ almost surely, so their sum lies almost surely in the closed set $\overline{S + S}$ . Scaling by $\sqrt{2}$ is a homeomorphism, so step 1.1 gives $\sqrt{2} S = \overline{S + S}$ . Taking any $s \in S$ and $- s \in S$ shows $0 \in \sqrt{2} S$ , hence $0 \in S$ . Now $x + 0 \in \overline{S + S} = \sqrt{2} S$ gives $S \subseteq \sqrt{2} S$ , while $x + x \in \sqrt{2} S$ gives $\sqrt{2} x \in S$ and hence $\sqrt{2} S \subseteq S$ . Therefore $\sqrt{2} S = S$ and $\overline{S + S} = S$ . The set $S$ is additive and symmetric, and division by $2$ follows by applying division by $\sqrt{2}$ twice. It is closed under dyadic rational scalars, and closure gives all real scalars. Thus $S$ is a closed linear subspace.

2.2A1F4F5F7F8F9step 1.1constructalgebra

Put $Z = B_{1}$ and $M = \max_{0 \leq t \leq 1} β_{t}$ . Every finite vector of bridge values is jointly Gaussian with $Z$ and has zero cross-covariance, because $Cov (B_{t} - t B_{1}, B_{1}) = t - t = 0$ . By [F8] these finite vectors are independent of $Z$ . The coordinate generation in step 1.1 and finite-measure uniqueness [F4], applied to rectangles and then the generated sigma-algebra, show that the entire bridge path, and hence $M$ , is independent of $Z$ . Write $ϕ$ for the standard-normal density. For $c > ε > 0$ , [F9] implies that the measure $P (\max_{t \leq 1} B_{t} < c, Z \in d z)$ on $[- ε, ε]$ has density $ϕ (z) - ϕ (2 c - z)$ : subtract the reflected-event measure from the terminal normal measure; the reflected-event CDF in [F9] differentiates to $ϕ (2 c - z)$ for $z < c$ . Thus $A_{ε} (c) : = \frac{P (\max_{t \leq 1} B_{t} < c, ∣ Z ∣ \leq ε)}{P (∣ Z ∣ \leq ε)} = \frac{\int_{- ε}^{ε} [ϕ (z) - ϕ (2 c - z)] d z}{\int_{- ε}^{ε} ϕ (z) d z} .$ For fixed $a > 0$ , take $0 < ε < a / 3$ . Since $∥ B - β ∥_{\infty} \leq ε$ on ${∣ Z ∣ \leq ε}$ , the inclusions with thresholds $a - ε$ and $a + ε$ , together with independence, give $A_{ε} (a - ε) \leq P (M < a) \leq A_{ε} (a + ε)$ . Continuity and positivity of $ϕ$ show that both bounds tend to $1 - ϕ (2 a) / ϕ (0) = 1 - e^{- 2 a^{2}}$ . Hence $P (M < a) = 1 - e^{- 2 a^{2}}$ for every $a > 0$ . By [F5], $P (M \leq a) = \lim_{k \to \infty} P (M < a + 1 / k) = 1 - e^{- 2 a^{2}}$ for $a > 0$ , and $M \geq 0$ together with the same formula as $a ↓ 0$ gives $P (M = 0) = 0$ . Therefore $M$ is atomless.

3.1F3step 2.1givenalgebra

If $X$ has any nondegenerate scalar coordinate, then $P (X = 0) = 0$ and $S \neq {0}$ . For every $r > 0$ , $0 \in S$ gives $P (∥ X ∥ < r) > 0$ . A nonzero $s \in S$ can be scaled to have norm greater than $r$ , so some neighborhood of that scaled point lies in ${∥ X ∥ > r}$ and has positive probability. Likewise, for $0 < a < b$ , scale $s$ to have norm $(a + b) / 2$ ; a sufficiently small ball about it is contained in ${a < ∥ X ∥ < b}$ and has positive probability. Consequently its norm CDF is strictly between zero and one at every positive argument and strictly increasing there. This applies to $G$ : any nonconstant wavelet has positive $P_{0}$ -variance, because $f_{0} \geq ρ$ makes the $P_{0}$ and Lebesgue null sets identical and a nonconstant wavelet cannot equal a constant almost everywhere.

4.1F3F5step 3.1givenalgebra

For each nonzero $x \in M_{0} (w)$ , its norm is attained by some coordinate: if $r = ∥ x ∥_{M (w)} > 0$ , the normalized level maxima are below $r / 2$ after finitely many levels, and the maximum over the remaining finite set equals $r$ . Enumerate all coordinates by $j$ , and write $b_{j} > 0$ for their norm weights: $b_{j} = w_{l}$ on detail level $l$ , and $b_{j} = 1$ for any additional Haar scaling coordinate. Hence, for $r > 0$ , ${∥ G ∥_{M (w)} = r}$ is contained in the countable union of events ${∣ G_{j} ∣ = r b_{j}}$ . Each has probability zero: a positive-variance normal has a density, and a zero-variance centered normal is zero whereas $r b_{j} > 0$ . By [F5] the union is null. At $r = 0$ , the nondegenerate coordinate from step 3.1 again gives probability zero; negative values are impossible. Absence of atoms and continuity of probability in [F5] show that this norm CDF is continuous on $R$ .

4.2A1F1F2F3F6F7step 1.1step 3.1givenalgebra

First verify the bridge used in [F7] directly. Every finite linear combination of $β_{t_{j}} = B_{t_{j}} - t_{j} B_{1}$ is a finite linear combination of Brownian values with time $1$ appended, so it is normal by [F7], including repeated times and singular combinations. Finite expectation linearity makes its mean zero. Covariance bilinearity gives $Cov (β_{s}, β_{t}) = \min (s, t) - t s - s t + s t = \min (s, t) - s t$ for $s, t \in [0, 1]$ . On the single continuous zero-start Brownian path event supplied in [F7], subtraction of the continuous function $t B_{1}$ makes $β$ continuous, with $β_{0} = β_{1} = 0$ . Thus it is the needed centered continuous Gaussian bridge, without an example as a logical supplier. Write $Y = J G$ . For every continuous real linear functional $L$ on $C ([0, 1])$ , $L \circ J$ is continuous linear on $M_{0} (w)$ , so $L (Y)$ is centered normal by [F3]; thus $Y$ is centered Gaussian. For $s \in [0, 1]$ , let $p_{N} 1_{[0, s]}$ be the orthogonal wavelet projection through level $N$ . Its Lebesgue $L^{2}$ limit is $1_{[0, s]}$ by [F2], and $f_{0} \leq D$ gives the same convergence in $L^{2} (P_{0})$ . Put $U_{N} (s) = J (P_{N} G) (s)$ . The coefficient covariance in [F3] gives $Cov (U_{N} (s), U_{M} (t)) = P_{0} [(p_{N} 1_{[0, s]} - P_{0} p_{N} 1_{[0, s]}) (p_{M} 1_{[0, t]} - P_{0} p_{M} 1_{[0, t]})]$ . In particular $U_{N} (s)$ is Cauchy in $L^{2}$ , and it converges almost surely to $Y (s)$ by [F1]. Its $L^{2}$ limit therefore equals $Y (s)$ almost surely, since convergence in $L^{2}$ implies convergence in probability and two probability limits agree. The means are zero; applying [F6] to the product differences passes the covariances to the limit. Thus $Y$ is a centered continuous Gaussian process with covariance $F_{0} (s \land t) - F_{0} (s) F_{0} (t)$ . The function $F_{0}$ is a continuous strictly increasing bijection of $[0, 1]$ onto itself. By [F3], $Y$ and $t \mapsto β_{F_{0} (t)}$ from [F7] have the same finite Gaussian laws; step 1.1 then gives equality of their Borel laws on $C ([0, 1])$ . In particular $∥ Y ∥_{\infty} \overset{d}{=} ∥ β ∥_{\infty}$ , and $Y$ has a nondegenerate evaluation coordinate, for example at any $s$ with $0 < F_{0} (s) < 1$ . Step 3.1 therefore applies to its norm as well.

5.1F3F5step 4.2step 2.2algebra

By symmetry of the centered Gaussian bridge, $- \min_{t} β_{t}$ has the same atomless distribution as $M$ . For $r \geq 0$ , continuity on a compact interval gives ${∥ β ∥_{\infty} = r} \subseteq {M = r} \cup {- \min_{t} β_{t} = r}$ . Both events on the right are null by step 2.2 and symmetry; [F5] gives no atom at $r$ . Negative norm values are impossible. By step 4.2, the CDF of $∥ J G ∥_{\infty}$ is therefore continuous on $R$ .

6.1F1F5step 3.1step 4.1step 4.2step 5.1givenalgebra∎

Both norm CDFs are continuous by steps 4.1 and 5.1, vanish at zero, and are strictly increasing on $(0, \infty)$ with values strictly between zero and one by steps 3.1 and 4.2. They tend to one at infinity because both random norms are finite almost surely. For each $u \in (0, 1)$ , continuity therefore gives a positive finite point $q$ with CDF value $u$ , strict increase gives its uniqueness, and monotonicity makes it the lower $u$ -quantile. Any sufficiently small open interval around $q$ is contained in $(0, \infty)$ , where strict increase holds. These are all the asserted quantile conditions.

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Finite Dirichlet gamma representation and weighted Bernstein bound

Statement

Assume AC. Let $β_{1}, \dots, β_{m} > 0$ , $B = \sum_{j} β_{j}$ , and let $Z_{j}$ be independent $Gamma (β_{j}, 1)$ variables. Put $S = \sum_{j} Z_{j}$ . Then $ω = (Z_{j} / S)_{j}$ is $Dirichlet (β_{1}, \dots, β_{m})$ , and $S$ is $Gamma (B, 1)$ , with mean $B$ and variance $B$ . Each $Z_{j}$ has mean $β_{j}$ , variance $β_{j}$ , and moment-generating function $E e^{u Z_{j}} = (1 - u)^{- β_{j}}$ for every real $u < 1$ . For real $q_{j}$ satisfying $∣ q_{j} ∣ \leq b$ , put $T (q) = \sum_{j} q_{j} (Z_{j} - β_{j})$ and $V = \sum_{j} β_{j} q_{j}^{2}$ . If $V > 0$ , then for $x > 0$ , $\Pr {∣ T (q) ∣ > x} \leq 2 \exp {- \frac{x^{2}}{2 (V + b x)}} .$ If $V = 0$ , $T (q) = 0$ almost surely. Sums of disjoint groups of the $Z_{j}$ are independent Gamma variables with their respective summed shapes.

Facts & Assumptions

Given: Full AC, the positive finite shape vector, and the weights in the Statement.

[F1]

Shape-rate Gamma distribution and The shape-rate Gamma density is normalized supply the normalized density $z^{a - 1} e^{- z} / Γ (a)$ , $z > 0$ , for $a > 0$ .

[F2]

Dirichlet distribution, including the one-category case specifies the normalized density $Γ (B) \prod_{j} ω_{j}^{β_{j} - 1} / \prod_{j} Γ (β_{j})$ on the simplex, and the point mass at one for $m = 1$ .

[F3]

Finite product laws and nonnegative changes of variables are supplied by For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique and A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions.

[F4]

Markov's inequality for random variables: for a nonnegative random variable $Y$ and $a > 0$ , $\Pr (Y \geq a) \leq E Y / a$ . Applied to $Y = e^{λ T (q)}$ with $λ > 0$ and $a = e^{λ x}$ , it gives $\Pr (T (q) \geq x) \leq e^{- λ x} E e^{λ T (q)}$ .

[F5]

Uniqueness of a multivariate exponential transform identifies laws having the same finite moment-generating transform near zero under AC.

Proof

Given: The hypotheses in the Statement.

1.1F1F2F3givenalgebracases

The product Gamma law exists by [F3]. In its density make the substitution $z_{j} = s ω_{j}$ , $ω_{m} = 1 - \sum_{j < m} ω_{j}$ . Its Jacobian is $s^{m - 1}$ : factor $s$ from the proportion columns of the derivative matrix and add its first $m - 1$ rows to the last; the remaining last row has first entry one and all other entries zero. The resulting density factors as $s^{B - 1} e^{- s} / Γ (B)$ times the simplex density in [F2]. Integrating either normalized factor proves both distributional assertions. For $m = 1$ , $ω_{1} = 1$ directly.

2.1F1F3F5step 1.1algebra

Substituting $y = (1 - u) z$ in the Euler Gamma density gives $E e^{u Z_{j}} = (1 - u)^{- β_{j}}$ for $u < 1$ . Its first two derivatives at zero yield mean $β_{j}$ and variance $β_{j}$ ; differentiation is justified by an exponential majorant with any fixed parameter in $(0, 1)$ . Independence makes the joint transform of disjoint group sums the product $\prod_{h} (1 - u_{h})^{- \sum_{j \in I_{h}} β_{j}}$ , which is the transform of the corresponding product Gamma law. By [F5], these are the same laws. In particular $S$ has the asserted mean and variance, also following from step 1.1.

3.1F4step 2.1algebracases∎

For $∣ λ ∣ b < 1$ , independence and step 2.1 give $\log E e^{λ T (q)} = \sum_{j} β_{j} [- \log (1 - λ q_{j}) - λ q_{j}] \leq λ^{2} V / [2 (1 - ∣ λ ∣ b)]$ . Indeed the convergent logarithm series gives $- \log (1 - u) - u \leq \sum_{r \geq 2} ∣ u ∣^{r} / r \leq u^{2} / [2 (1 - ∣ u ∣)]$ for $∣ u ∣ < 1$ . If $V > 0$ , set $λ = x / (V + b x)$ ; exponential Markov then bounds the upper tail by $\exp [- x^{2} / (2 (V + b x))]$ . Apply the same calculation to $- q$ and sum the bounds. If $V = 0$ , positivity of all $β_{j}$ forces every $q_{j} = 0$ , so $T (q) = 0$ .

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Type-H histogram posterior weak Bernstein–von Mises theorem

Statement

Assume AC. Use the type-H prior of Type-S log-density and type-H histogram prior families, retaining its original conditions: $1 / 2 < α \leq 1$ , $f_{0} \in C^{α} ([0, 1])$ , $0 < ρ_{0} \leq f_{0} \leq D_{0} < \infty$ , either original cutoff $L_{n} = j_{n}$ or $L_{n} = ℓ_{n}$ , and $c_{1} 2^{- L_{n} a} \leq a_{k, L_{n}} \leq c_{2}$ for fixed positive $a, c_{1}, c_{2}$ . Write $m_{n} = 2^{L_{n}}$ . In the Haar convention, the scaling function is $1$ , the detail functions are $ψ_{l k} = 2^{l / 2} (1_{I_{l + 1, 2 k}} - 1_{I_{l + 1, 2 k + 1}}), l \geq 0, 0 \leq k < 2^{l},$ and a histogram at resolution $L_{n}$ has only the scaling coordinate and levels $l < L_{n}$ . Let $X_{1}, \dots, X_{n}$ be IID from $f_{0}$ , let $N_{n k} = \sum_{i} 1_{I_{L_{n}, k}} (X_{i})$ , and define the empirical histogram $h_{n} (x) = m_{n} N_{n k} / n on I_{L_{n}, k} .$ Let $Π_{n} (\cdot ∣ X^{n})$ be the density posterior for the stated histogram prior. For every nondecreasing $w_{l} \geq 1$ with $w_{l} / \sqrt{l + 1} \to \infty$ , its conditional law $Q_{n}$ of $\sqrt{n} (f - h_{n})$ satisfies $β_{B L, M_{0} (w)} (Q_{n}, L (G_{P_{0}})) ⟶ 0$ in $P_{0}^{n}$ -probability, where the Borel centered Gaussian bridge $G_{P_{0}}$ is supplied by A bounded-density Gaussian bridge has a Borel multiscale law and has covariance $P_{0} [(ψ - P_{0} ψ) (ψ^{'} - P_{0} ψ^{'})]$ . Its scaling coordinate is zero. This proves the type-H source-specific posterior BvM with the empirical projection center; no additional contraction, bias, or Dirichlet-parameter condition is imposed.

Facts & Assumptions

Given: AC, the original type-H truth, cutoff, parameter, Haar, observation, and admissible-weight conditions in the Statement.

[F1]

Finite Dirichlet gamma representation and weighted Bernstein bound: a Dirichlet vector of shapes $β_{k}$ is $Z_{k} / S$ for independent Gamma $(β_{k}, 1)$ variables; $S$ has mean and variance $B = \sum_{k} β_{k}$ . For $T (q) = \sum_{k} q_{k} (Z_{k} - β_{k})$ , $∣ q_{k} ∣ \leq b$ , and $V = \sum_{k} β_{k} q_{k}^{2} > 0$ , $\Pr (∣ T (q) ∣ > x) \leq 2 e^{- x^{2} / [2 (V + b x)]}$ . Disjoint Gamma shapes add, and $E e^{u Z_{k}} = (1 - u)^{- β_{k}}$ for $u < 1$ .

[F2]

Dirichlet-Multinomial conjugate update and Bayes formula for dominated kernels identify the positive finite normalized Dirichlet likelihood update.

[F3]

A bounded-density Gaussian bridge has a Borel multiscale law supplies the Borel Gaussian bridge law for the bounded density in the Statement.

[F4]

Finite-projection convergence and posterior tail control imply weak BvM concludes BL convergence from every fixed finite-coordinate BL limit and vanishing posterior tails in the iterated limit.

[F5]

Prokhorov tightness theorem on polish spaces gives subsequential weak compactness of tight finite-dimensional Borel laws under AC; Uniqueness of a multivariate exponential transform identifies finite laws with a common moment-generating transform near zero.

[F6]

Vector Slutsky theorem passes a finite-dimensional weak limit through algebraic operations with quantities converging in probability to constants. The bounded-Lipschitz metric characterizes weak convergence on Polish spaces converts finite-dimensional weak convergence to BL convergence.

[F7]

Markov's inequality for random variables and Finite and countable subadditivity of measures give moment, exponential, and union bounds.

[F8]

Finite type-S and type-H prior draws are L2 probability densities supplies the finite histogram density construction. Little-o multiscale coefficient space M0(w) supplies the coefficient norm and admissibility convention, with the finite scaling coordinate included.

Proof

Given: All hypotheses in the Statement, with full AC throughout.

1.1F1F2givenalgebra

Abbreviate $m = m_{n}$ , $a_{k} = a_{k, L_{n}}$ , $A = \sum_{k} a_{k}$ , $β_{k} = N_{n k} + a_{k}$ , and $B = n + A$ . The original cutoff conditions and $α > 1 / 2$ give $m / \sqrt{n} \to 0$ and $m \log n / n \to 0$ , and $A \leq c_{2} m$ . On the positive simplex, the full-data histogram likelihood is $m^{n} \prod_{k} ω_{k}^{N_{n k}}$ . Multiplication by the prior density yields a constant times $\prod_{k} ω_{k}^{β_{k} - 1}$ , with every shape positive; its normalization is finite positive by [F2]. Therefore the full-data posterior is Dirichlet $(β_{k})_{k}$ , regardless of whether the actual sampling truth is itself a histogram. By [F1] realize it by $ω_{k} = Z_{k} / S$ . Write $R_{n}$ for the posterior mean histogram measure, identified with its density in coefficient differences, with cell masses $β_{k} / B$ . For a cell-constant function $q$ , set $T (q) = \sum_{k} q_{k} (Z_{k} - β_{k})$ . Direct subtraction gives $\sqrt{n} (ω q - R_{n} q) = (\sqrt{n} / S) T (q - R_{n} q)$ .

2.1F7step 1.1algebra

Fix constants $t > 0$ and $C$ large enough that $c : = t C - D_{0} (e^{t} - 1) > 0$ . Since $N_{n k}$ is Binomial $(n, p_{k})$ with $p_{k} = P_{0} I_{L_{n}, k} \leq D_{0} / m$ , independence of the observations gives $E e^{t N_{n k}} = (1 + p_{k} (e^{t} - 1))^{n} \leq e^{n D_{0} (e^{t} - 1) / m}$ . Hence $E_{n} = {\max_{k} N_{n k} \leq C n / m}$ satisfies $P_{0}^{n} (E_{n}^{c}) \leq m e^{- c n / m} \to 0$ , by exponential Markov and the union bound. On $E_{n}$ , for all sufficiently large $n$ , $β_{k} \leq C^{'} n / m$ , $B \leq 2 n$ , and both the empirical histogram density and the mean histogram density are bounded by a fixed constant $C^{'}$ independent of $n$ .

2.2F1F7step 1.1givenalgebra

Fix finitely many Haar coordinates. Choose a fixed coarse dyadic partition of $r$ cells on which all their functions are constant. For large $n$ its cells are unions of level- $L_{n}$ cells. By [F1], the corresponding grouped Gamma variables $Z_{n j}$ are independent Gamma $(b_{n j}, 1)$ with $b_{n j} = N_{n} (J_{j}) + \sum_{I_{L_{n}, k} \subset J_{j}} a_{k}$ . Their summed shape is still $B$ . Write $p_{j} = P_{0} (J_{j}) > 0$ . Each coarse count satisfies $E [N_{n} (J_{j}) / n] = p_{j}$ and variance at most $1 / (4 n)$ , by observation independence. Markov applied to squared deviations, together with $A / n \to 0$ , proves $b_{n j} / n \to p_{j}$ in $P_{0}^{n}$ -probability simultaneously for the fixed finite partition.

3.1F1F7step 1.1step 2.1algebracases

On $E_{n}$ , take any Haar detail $ψ = ψ_{l k}$ with $l < L_{n}$ and put $q = ψ - R_{n} ψ$ . Then $∣ q ∣ \leq 2 2^{l / 2}$ and $V = \sum_{k} β_{k} q_{k}^{2} = B {Var}_{R_{n}} (ψ) \leq B R_{n} ψ^{2} \leq C^{'} n$ , using $\int ψ^{2} = 1$ and the density bound. By [F1] and [F7], $\Pr (S < B / 2 ∣ X^{n}) \leq 4 / B$ , because $Var (S) = B$ . On the complement, step 1.1 yields $∣ \sqrt{n} (ω ψ - R_{n} ψ) ∣ \leq 2 ∣ T (q) ∣ / \sqrt{n}$ . Thus, apart from the single event ${S < B / 2}$ , the conditional probability that this absolute coefficient exceeds $x$ is at most $2 \exp {- c_{0} x^{2} / (1 + 2^{l / 2} x / \sqrt{n})}$ for a fixed $c_{0} > 0$ . The zero-variance case has no such failure event, by [F1].

3.2F1F5F7step 2.2algebra

Consider any deterministic data sequence along which $b_{n j} / n \to p_{j}$ for every $j$ . Under its conditional Gamma product laws put $Y_{n j} = (Z_{n j} - b_{n j}) / \sqrt{n}$ . For every fixed $t \in R^{r}$ , [F1] and the logarithm series give $\log E e^{t \cdot Y_{n}} = \sum_{j} b_{n j} [- \log (1 - t_{j} / \sqrt{n}) - t_{j} / \sqrt{n}] \to \frac{1}{2} \sum_{j} p_{j} t_{j}^{2}$ . The remainder is $O (n^{- 3 / 2} \sum_{j} b_{n j} ∣ t_{j} ∣^{3}) = O (n^{- 1 / 2})$ . The means of $Y_{n}$ are zero and their total second moments are $B / n = O (1)$ , so their laws are tight by the squared-norm Markov bound and compactness of finite-dimensional closed balls. By [F5], any subsequence has a further weak limit. For any fixed $t$ , the same expansion at $2 t$ uniformly bounds $E e^{2 t \cdot Y_{n}}$ ; hence $e^{t \cdot Y_{n}}$ is uniformly integrable. Truncation of this continuous exponential therefore passes its expectation through each weakly convergent subsequence. The limit law has transform $\exp (\frac{1}{2} \sum_{j} p_{j} t_{j}^{2})$ and an exponential moment on a neighborhood of zero: $e^{δ \sum_{j} ∣ y_{j} ∣} \leq \sum_{ϵ \in {- 1, 1}^{r}} e^{δ ϵ \cdot y}$ and the corresponding transforms are finite. The independent centered normal vector with variances $p_{j}$ has this same transform, by completing the square in each normal density. Transform uniqueness [F5] identifies every subsequential limit, proving $Y_{n} \Rightarrow N_{r} (0, diag (p))$ .

4.1F4F7F8step 2.1step 3.1algebra

Choose a fixed $K > 0$ sufficiently large. Uniformly over $l < L_{n}$ , $2^{l / 2} K \sqrt{l + 1} / \sqrt{n} \leq K \sqrt{m (L_{n} + 1) / n} \to 0$ . Summing the exponential bounds of step 3.1 over levels $J < l < L_{n}$ and their $2^{l}$ coefficients therefore gives, for all large $n$ , $Π_{n} {\max_{J < l < L_{n}, k} ∣ \sqrt{n} (f - R_{n})_{l k} ∣ / \sqrt{l + 1} > K ∣ X^{n}} \leq 4 / B + \sum_{l > J} 2^{l + 1} e^{- c_{1} K^{2} (l + 1)}$ on $E_{n}$ , with fixed $c_{1} > 0$ . The probability $4 / B$ is counted only once, since the same $S$ normalizes every coordinate. Choose $K$ with $c_{1} K^{2} > 2 \log 2$ ; the displayed series tail then tends to zero with $J$ . For any $ε > 0$ , admissibility gives $K \sqrt{l + 1} \leq ε w_{l}$ for all sufficiently large $l$ . Both $f$ and $R_{n}$ are histograms, so all their detail coordinates at $l \geq L_{n}$ vanish. Consequently their conditional $M (w)$ tails vanish in the iterated limit of [F4], first $n \to \infty$ , then $J \to \infty$ , since $P_{0}^{n} (E_{n}^{c}) \to 0$ .

5.1F8step 1.1step 2.1step 4.1algebra

For a Haar detail $ψ = ψ_{l k}$ , exact subtraction gives $\sqrt{n} (R_{n} ψ - P_{n} ψ) = \sqrt{n} [\sum_{k} a_{k} ψ_{k} - A P_{n} ψ] / B$ . Haar cell values satisfy $m^{- 1} \sum_{k} ∣ ψ_{k} ∣ = \int ∣ ψ ∣ = 2^{- l / 2} \leq 1$ . Hence $∣ \sum_{k} a_{k} ψ_{k} ∣ \leq c_{2} m$ , while on $E_{n}$ , $∣ P_{n} ψ ∣ \leq C \int ∣ ψ ∣ \leq C$ . This bounds every such mean shift by $C^{''} m / \sqrt{n} \to 0$ . The scaling-coordinate shift is exactly zero, and all coordinates at $l \geq L_{n}$ vanish for both centers. Since $w_{l} \geq 1$ , $∥ \sqrt{n} (R_{n} - h_{n}) ∥_{M (w)} \to 0$ uniformly on $E_{n}$ . In particular step 4.1 also supplies the required posterior tails about $h_{n}$ .

6.1F3F6step 5.1step 3.2algebra

Here $(S - B) / \sqrt{n} = \sum_{j} Y_{n j}$ , $S / n \to 1$ in conditional probability by its variance $B / n^{2} \to 0$ and mean $B / n \to 1$ . Exact algebra gives $\sqrt{n} (Z_{n j} / S - b_{n j} / B) = (n / S) [Y_{n j} - (b_{n j} / B) \sum_{h} Y_{n h}]$ . By [F6] and step 3.2, this vector converges to $(U_{j} - p_{j} \sum_{h} U_{h})_{j}$ , with $U_{j}$ independent $N (0, p_{j})$ . Its covariance is $diag (p) - p p^{T}$ . Applying the fixed linear map from coarse-cell masses to the chosen Haar coordinates yields covariance exactly $P_{0} [(ψ - P_{0} ψ) (ψ^{'} - P_{0} ψ^{'})]$ . Step 5.1 removes the mean shift from $R_{n}$ to $h_{n}$ on $E_{n}$ . Thus along every deterministic data sequence with the ratios in step 2.2 converging and $E_{n}$ eventually occurring, the finite-coordinate posterior converges weakly, hence in BL by [F6], to the corresponding projection of [F3].

7.1F6step 2.1step 2.2step 6.1givenchoosecontradiction

The preceding deterministic-sequence conclusion implies the finite-coordinate BL convergence in $P_{0}^{n}$ -probability. To see this without a simultaneous conditional null-set assertion, if that convergence failed, there would be $ε, η > 0$ and infinitely many $n$ with probability at least $η$ that the distance exceeds $ε$ . Along a subsequence, step 2.2 and $P_{0}^{n} (E_{n}^{c}) \to 0$ make the probability of a ratio error above $1 / q$ or $E_{n}^{c}$ smaller than $η / 2$ at its $q$ th chosen index. Thus at each such index there exists data with distance above $ε$ , ratio error below $1 / q$ , and $E_{n}$ true. AC selects a deterministic sequence of those data, contradicting step 6.1. This proves the asserted convergence in probability for each fixed finite projection.

8.1F3F4F8step 4.1step 5.1step 7.1algebra∎

The posterior histogram construction [F8] and its finite-dimensional simplex kernel make $Q_{n}$ a data-indexed Borel probability on $M_{0} (w)$ : each draw and $h_{n}$ has only finitely many Haar coefficients, and its map from simplex coordinates is continuous. The target is the Borel bridge law [F3]. Step 7.1 supplies every fixed finite-projection BL limit, and steps 4.1, 5.1 supply its posterior-tail condition. Apply [F4] to obtain the stated $M_{0} (w)$ BL convergence. Every original type-H assumption has been retained; in particular the lower parameter bound is compatible with, but is not needed beyond, positivity in this conjugate proof.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Compact wavelet projections approximate smooth functions and commute asymptotically with multiplication

Statement

Let $V_{L}$ be the span of wavelets through level $L$ in a compactly supported orthonormal multiresolution basis on $[0, 1]$ , either boundary corrected or periodized. Assume constants belong to $V_{L}$ , its dimension $d_{L} ≍ 2^{L}$ , and its orthonormal scaling functions $ϕ_{L k}$ have supremum norm at most $C 2^{L / 2}$ , support diameter at most $C 2^{- L}$ , and uniformly bounded overlap. Periodized supports and distances are interpreted on the circle. Assume also $\sum_{k} ∣ ψ_{l k} (x) ∣ \leq C 2^{l / 2}$ and $∥ ψ_{l k} ∥_{1} \leq C 2^{- l / 2}$ . Let $P_{L}$ denote the orthogonal projection onto $V_{L}$ .

Then $P_{L}$ has a kernel $K_{L}$ satisfying $\sup_{x} \int ∣ K_{L} (x, y) ∣ d y \leq C^{'}$ and $K_{L} (x, y) = 0$ when $dist (x, y) > C^{'} 2^{- L}$ . If $a$ is $β$ -Hölder for $0 < β \leq 1$ , then $∥ P_{L} (a h) - a h ∥_{\infty} \leq C^{'} ∥ a ∥_{C^{β}} 2^{- β L} ∥ h ∥_{\infty}, h \in V_{L} .$ For the composition assertion, assume additionally an integer $m > α$ such that $∥ ψ_{l k}^{(q)} ∥_{\infty} \leq C_{q} 2^{l (q + 1 / 2)}$ for $0 \leq q \leq m$ , bounded overlap also for their derivatives, and vanishing moments through degree $m - 1$ for the detail wavelets, including boundary wavelets; the finite low scaling functions are $C^{m}$ . These are the usual sufficiently regular compact wavelet hypotheses. If a bounded function $g$ has coefficients bounded by $C_{g} 2^{- l (α + 1 / 2)}$ , with $α > 0$ , and its wavelet expansion represents $g$ in $L^{2}$ , then $∥ g - P_{L} g ∥_{\infty} \leq C^{''} C_{g} 2^{- α L}$ , up to equality of the continuous representatives. Under these additional derivative and cancellation hypotheses, $e^{g}$ also has coefficients bounded by $C_{g}^{'} 2^{- l (α + 1 / 2)}$ , including integer $α$ , and $g, e^{g}$ are $β$ -Hölder for every $0 < β < \min (α, 1)$ . Moreover $∥ h ∥_{\infty} \leq C^{''} \sqrt{d_{L}} ∥ h ∥_{2}$ for $h \in V_{L}$ .

Facts & Assumptions

Given: The explicitly stated compact multiresolution, scaling-function, wavelet, and Hölder assumptions.

[F1]

Logarithmic negative Sobolev spaces for white noise supplies the orthonormal coefficient convention; all additional localization hypotheses are stated above.

Proof

1.1F1givenalgebra

The finite orthogonal projection has kernel $K_{L} (x, y) = \sum_{k} ϕ_{L k} (x) ϕ_{L k} (y)$ . At each $x$ only a uniformly bounded number of terms are nonzero, and each has $∥ ϕ_{L k} ∥_{1} \leq C 2^{- L / 2}$ . Thus $\int ∣ K_{L} (x, y) ∣ d y \leq \sum_{k : ϕ_{L k} (x) \neq 0} ∣ ϕ_{L k} (x) ∣ ∥ ϕ_{L k} ∥_{1} \leq C^{'}$ . A nonzero term requires $x, y$ to lie in one scaling support, giving the support bound. Also $K_{L} (x, x) = \sum_{k} ϕ_{L k} (x)^{2} \leq C^{'} 2^{L}$ ; Cauchy–Schwarz on the scaling coefficients gives the claimed $L^{2}$ to $L^{\infty}$ bound.

1.2F1givenalgebra

The levelwise tail satisfies $∥ \sum_{k} ⟨ g, ψ_{l k} ⟩ ψ_{l k} ∥_{\infty} \leq C C_{g} 2^{- α l}$ . Summing its geometric series over $l > L$ proves uniform convergence and the claimed approximation rate. Its uniform limit agrees with its $L^{2}$ limit, hence with $g$ almost everywhere, and everywhere for continuous representatives.

2.1givenstep 1.1algebra

For $h \in V_{L}$ , $P_{L} h = h$ . Therefore $P_{L} (a h) (x) - a (x) h (x) = \int K_{L} (x, y) (a (y) - a (x)) h (y) d y$ . Hölder continuity, the support bound, and the kernel integral bound give the asserted commutator estimate.

2.2givenstep 1.2algebra

For composition put $u_{j} = P_{j} g$ and $v_{j} = u_{j} - u_{j - 1}$ , with the first finite low block treated as $u_{0}$ . Levelwise derivative bounds and overlap give $∥ v_{j}^{(q)} ∥_{\infty} \leq C_{g} 2^{j (q - α)}$ for $0 \leq q \leq m$ , and $∥ u_{j}^{(q)} ∥_{\infty} \leq C_{g}^{'} 2^{j q}$ for $1 \leq q \leq m$ by summing levels; also $\sup_{j} ∥ u_{j} ∥_{\infty} < \infty$ . The identity $e^{u_{j}} - e^{u_{j - 1}} = v_{j} \int_{0}^{1} e^{u_{j - 1} + t v_{j}} d t$ and repeated product and chain rules therefore give, writing $H_{j} = e^{u_{j}} - e^{u_{j - 1}}$ , $∥ H_{j} ∥_{\infty} \leq C 2^{- j α}$ and $∥ H_{j}^{(m)} ∥_{\infty} \leq C 2^{j (m - α)}$ . Indeed a derivative of order $q$ of the exponential factor is a finite sum of products of derivatives of its exponent of total order $q$ , bounded by $C 2^{j q}$ ; pairing with the derivative of order $m - q$ of $v_{j}$ gives $C 2^{j (m - α)}$ .

3.1givenstep 2.2algebra

The telescoping series $e^{g} = e^{u_{0}} + \sum_{j \geq 1} H_{j}$ converges uniformly. For a wavelet at level $l$ , the terms with $j \geq l$ have coefficients bounded by $C 2^{- l / 2} \sum_{j \geq l} 2^{- j α} \leq C 2^{- l (α + 1 / 2)}$ . For $j < l$ , subtract the degree $m - 1$ Taylor polynomial of $H_{j}$ about a point of the wavelet support; all its integrals vanish by the stated moments. The Taylor remainder gives $∣ ⟨ H_{j}, ψ_{l k} ⟩ ∣ \leq C 2^{- l (m + 1 / 2)} 2^{j (m - α)}$ . Sum the geometric series over $j < l$ to get the same $C 2^{- l (α + 1 / 2)}$ bound. The fixed smooth $e^{u_{0}}$ has the stronger order $2^{- l (m + 1 / 2)}$ . This proves composition in the coefficient class directly, for every positive $α$ including integers, without presuming a Hölder–wavelet equivalence.

4.1givenstep 1.2step 2.2algebra∎

For $0 < β < \min (α, 1)$ the levelwise derivative and supremum bounds for $g$ are respectively at most $C 2^{j (1 - β)}$ and $C 2^{- j β}$ . Given distance $0 < δ < 1$ , choose $J$ with $2^{- J - 1} < δ \leq 2^{- J}$ . The difference of the levels through $J$ is at most $C δ \sum_{j \leq J} 2^{j (1 - β)} \leq C δ^{β}$ by the derivative bound; the remaining difference is at most $C \sum_{j > J} 2^{- j β} \leq C δ^{β}$ . Thus $g$ is $β$ -Hölder. The exponential has bounded derivative on the bounded range of $g$ , so $e^{g}$ has the same Hölder bound.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Type-S posterior contracts in quotient log-density coordinates without a tail restriction on a log-Lipschitz prior

Statement

Assume AC for the imported measure and Gaussian interfaces. Use Type-S priors and either cutoff of Type-S log-density and type-H histogram prior families with the compact multiresolution basis of Compact wavelet projections approximate smooth functions and commute asymptotically with multiplication, regular enough for the truth's Hölder approximation. Suppose $0 < ρ_{0} \leq f_{0} \leq D_{0}$ , $g_{0} = \log f_{0} \in C^{α}$ , and $α > 1 / 2$ . In S1 take $σ_{l} = 2^{- l (α + 1 / 2)}$ and any probability density whose logarithm is Lipschitz on all of $R$ . In S2 take Gaussian coefficients with $σ_{l} = 2^{- l (r + 1 / 2)}$ and $0 < r < α - 1 / 4$ .

Write $d = \dim V_{L_{n}} ≍ 2^{L_{n}}$ , $E_{L_{n}} = V_{L_{n}} \cap {h : \int h = 0}$ , $θ = T - \int T$ , $θ_{*} = P_{L_{n}} g_{0} - \int g_{0}$ , and $e_{n} = \sqrt{d / n} + d^{- α}$ . Under IID data of density $f_{0}$ , the posterior distributions of $∥ θ - θ_{*} ∥_{2} / e_{n}$ are tight in probability. Equivalently, for every deterministic $b_{n} \to \infty$ , $Π (∥ θ - θ_{*} ∥_{2} > b_{n} e_{n} ∣ X^{n}) \to 0$ in $P_{0}^{n}$ -probability. The normalizer defining the posterior is positive and finite $P_{0}^{n}$ -almost surely. The quotient prior has a positive continuous Lebesgue density $q_{L}$ on $E_{L}$ . In S1 it obeys $∣ \log q_{L} (θ + u) - \log q_{L} (θ) ∣ \leq B_{L} ∥ u ∥_{2}, B_{L}^{2} \leq C \sum_{l \leq L} 2^{l} σ_{l}^{- 2},$ and a shift in a single high wavelet coordinate obeys the sharper bound $C ∣ u_{l k} ∣ / σ_{l}$ . The low scaling block has a fixed finite collection of analogous weighted bounds. In S2 the quotient prior is centered Gaussian, with precision $Q_{L}$ diagonal on high wavelets with entries $σ_{l}^{- 2}$ and a fixed positive definite low scaling block. Its log-density increment is $- ⟨ Q_{L} θ, u ⟩ - ⟨ Q_{L} u, u ⟩ / 2$ .

Facts & Assumptions

Given: AC and the density, basis, cutoff, coefficient, and regularity hypotheses of the Statement.

[F1]

Compact wavelet projections approximate smooth functions and commute asymptotically with multiplication gives $∥ P_{L} g_{0} - g_{0} ∥_{\infty} \leq C d^{- α}$ and $∥ u ∥_{\infty} \leq C \sqrt{d} ∥ u ∥_{2}$ on $V_{L}$ .

[F2]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product permits integration of nonnegative likelihoods in either order.

[F3]

Markov's inequality for random variables bounds a nonnegative variable's upper tail by its expectation divided by the threshold.

Proof

1.1givenF2algebra

Quotient out constants before using likelihood curvature. Constants belong to the fixed low scaling space. An orthogonal change within that finite space separates $T = θ + c 1$ ; integrate $c$ out of the prior and write $q_{L}$ for the resulting Lebesgue density on $E_{L}$ . The likelihood depends only on $θ$ . In S1 its product density before integration obeys $p (θ + u, c) / p (θ, c) \in [e^{- B_{L} ∥ u ∥_{2}}, e^{B_{L} ∥ u ∥_{2}}]$ , where $B_{L}^{2} \leq C \sum_{l \leq L} 2^{l} σ_{l}^{- 2} \leq C d^{2 α + 2}$ . Integrating preserves both inequalities for $q_{L}$ ; it is positive, finite almost everywhere, and the log-Lipschitz inequality supplies its positive continuous finite version everywhere. In S2 the marginal is a nondegenerate Gaussian on $E_{L}$ , with precision $Q_{L}$ equal to $σ_{l}^{- 2}$ on high wavelets and a fixed positive definite low block; hence $∥ Q_{L} ∥ \leq C d^{2 r + 1}$ .

1.2F1F3givenalgebra

Let $f_{*} = e^{θ_{*}} / \int e^{θ_{*}}$ . [F1] gives $∥ f_{*} - f_{0} ∥_{\infty} \leq C d^{- α}$ . For $ℓ (θ) = n (P_{n} θ - \log \int e^{θ})$ , the centered score at $θ_{*}$ has expected squared Euclidean norm at most $C n d$ : sum the coordinate variances, each bounded by $D_{0}$ by orthonormality. The score's deterministic mean is the coefficient vector of $n (f_{0} - f_{*})$ , with norm at most $C n d^{- α}$ . Consequently [F3] gives $∥ \nabla ℓ (θ_{*}) ∥_{2} = O_{P} (n e_{n})$ . On every ball $∥ u ∥_{2} \leq R e_{n}$ with fixed $R$ , [F1] and $\sqrt{d} e_{n} \to 0$ make $f_{θ_{*} + u}$ bounded uniformly above and away from zero. For $v \in E_{L}$ , ${Var}_{f} (v) = \inf_{a} \int (v - a)^{2} f$ lies between $c ∥ v ∥_{2}^{2}$ and $C ∥ v ∥_{2}^{2}$ , using $\int v = 0$ . Thus $- \nabla^{2} ℓ$ is between $c n I$ and $C n I$ on these balls. Convexity of $- ℓ$ holds globally.

2.1givenstep 1.1algebra

In S1, $B_{L} \leq C n e_{n}$ by either cutoff. In S2 the Hölder coefficient bound gives $G_{L}^{2} : = ∥ Q_{L} θ_{*} ∥_{2}^{2} \leq C [1 + \sum_{l \leq L} 2^{l (4 r + 2 - 2 α)}]$ , so $G_{L} \leq C n e_{n}$ , and $∥ Q_{L} ∥ / n \to 0$ because $r < α$ . The Gaussian log-density increment is exactly $- ⟨ Q_{L} θ_{*}, u ⟩ - ⟨ Q_{L} u, u ⟩ / 2$ . In particular its radial upper slope is at most $G_{L}$ , and on the $e_{n}$ ball it is bounded below by $- G_{L} e_{n} - ∥ Q_{L} ∥ e_{n}^{2} / 2$ .

3.1step 1.1step 1.2step 2.1algebra

Fix a score event $∥ \nabla ℓ (θ_{*}) ∥_{2} \leq A n e_{n}$ . Put $r_{0} = R e_{n}$ , with $R$ a large fixed constant. Taylor's formula on the $r_{0}$ ball bounds $ℓ (θ_{*} + r_{0} ω) - ℓ (θ_{*})$ above by $A n e_{n} r_{0} - c n r_{0}^{2} / 2$ and its radial derivative above by $A n e_{n} - c n r_{0}$ . Global convexity of $- ℓ$ extends this derivative bound to every $r \geq r_{0}$ . Add the prior bound $B_{L} r$ in S1, or $G_{L} r$ with its nonpositive quadratic term in S2. For $R$ large enough the log unnormalized posterior ratio $H (r ω)$ relative to $θ_{*}$ is at most $- c^{'} n r_{0} r$ for all $r \geq r_{0}$ . On $r \leq e_{n}$ , the lower Taylor bound and the prior lower bounds instead give $H \geq - C_{A} n e_{n}^{2}$ .

4.1F3step 1.2step 3.1algebra

Polar integration in dimension $m = d - 1$ bounds the tail/normalizer ratio by $e^{C_{A} n e_{n}^{2}} m \int_{R}^{\infty} t^{m - 1} e^{- c^{'} n R e_{n}^{2} t} d t$ . Here $n e_{n}^{2} \geq d > m$ . Choose $R$ so large that $\log t \leq c^{'} R t / 4$ for $t \geq R$ and $C_{A} \leq c^{'} R^{2} / 4$ . The integral is then at most $[m / (c^{''} n R e_{n}^{2})] e^{- c^{''} n R^{2} e_{n}^{2}}$ after decreasing $c^{''} > 0$ , and tends to zero. The denominator used here is the integral on the $e_{n}$ ball, at least its volume times $e^{- C_{A} n e_{n}^{2}}$ times the positive unnormalized value at $θ_{*}$ . First increase $A$ to make its score event arbitrarily probable, then increase $R$ . This proves tightness. For any $b_{n} \to \infty$ , eventual containment of every fixed $R e_{n}$ ball in the $b_{n} e_{n}$ ball yields the stated equivalent formulation.

5.1F2givenalgebra∎

Every modeled density is everywhere positive. Its prior predictive product density is positive and integrates to one by [F2], so it is finite Lebesgue-almost everywhere on the sample cube. The law $P_{0}^{n}$ has a bounded Lebesgue density, hence the same exceptional set is $P_{0}^{n}$ -null. This proves finite positive evidence on the asserted fibers independently of the score event.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Type-S density posteriors satisfy the weak multiscale Bernstein–von Mises theorem

Statement

Assume AC. Let $X_{1}, \dots, X_{n}$ be IID on $[0, 1]$ with density $0 < ρ_{0} \leq f_{0} \leq D_{0} < \infty$ , and suppose $g_{0} = \log f_{0} \in C^{α}$ . Use a sufficiently regular compactly supported boundary corrected orthonormal multiresolution wavelet basis satisfying Compact wavelet projections approximate smooth functions and commute asymptotically with multiplication, with regularity greater than $α$ , and let $V_{L}$ contain exactly its wavelet levels through $L$ , including the fixed low scaling space and constants. Its regularity and vanishing moments satisfy the composition hypotheses in that lemma for some integer $m > α$ . Let $L_{n}$ be either of the two cutoff sequences in Type-S log-density and type-H histogram prior families, fixed consistently for all $n$ .

In Type-S1 take $α > 1$ , $σ_{l} = 2^{- l (α + 1 / 2)}$ , and any coefficient probability density with globally Lipschitz logarithm. In Type-S2 take $α > 1 / 2$ , Gaussian coefficients, and $σ_{l} = 2^{- l (r + 1 / 2)}$ with $0 < r < α - 1 / 4$ . Normalize $f = \exp (T) / \int \exp (T)$ as in that definition. For every admissible weight sequence $w$ , the posterior satisfies $β_{B L, M_{0} (w)} (L (\sqrt{n} (f - P_{n} (L_{n})) ∣ X^{n}), L (G_{P_{0}})) ⟶ 0$ in $P_{0}^{n}$ -probability, where $P_{n} (L_{n})$ has empirical wavelet coefficients through $L_{n}$ and zero coefficients above $L_{n}$ , and $G_{P_{0}}$ is the Borel Gaussian bridge with covariance $P_{0} [(ψ_{i} - P_{0} ψ_{i}) (ψ_{j} - P_{0} ψ_{j})]$ .

Facts & Assumptions

Given: The model, truth, multiresolution basis, prior scales, cutoff, admissible weights, and full AC in the Statement. Put $d = \dim V_{L_{n}} ≍ 2^{L_{n}}$ , $e_{n} = \sqrt{d / n} + d^{- α}$ , and $λ_{l} = \max (l, 1)$ to include the fixed low levels.

[F1]

Type-S posterior contracts in quotient log-density coordinates without a tail restriction on a log-Lipschitz prior gives quotient-coordinate contraction at radius $e_{n}$ around $θ_{*} = P_{L_{n}} g_{0} - \int g_{0}$ ; its Statement supplies the positive quotient marginal prior density and its shift bounds.

[F2]

Compact wavelet projections approximate smooth functions and commute asymptotically with multiplication gives the uniform projection kernel bound, approximation, $L^{2}$ to supremum bound, and the multiplication commutator estimate. Its composition conclusion proves that $f_{0} = e^{g_{0}}$ has the required $α$ coefficient decay, including integer $α$ , and that $f_{0}$ is $β$ -Hölder for every $0 < β < \min (α, 1)$ .

[F3]

A bounded-density Gaussian bridge has a Borel multiscale law constructs the Gaussian bridge law on $M_{0} (w)$ under AC.

[F4]

Finite-projection convergence and posterior tail control imply weak BvM reduces the asserted bounded-Lipschitz convergence to fixed finite projections and vanishing posterior tails.

[F5]

Bernstein tail bound for independent bounded sums gives Bernstein's inequality for independent centered bounded variables.

[F6]

Prokhorov tightness theorem on polish spaces, Uniqueness of a law from its characteristic function, and Cramer wold device give subsequential weak compactness from tightness, uniqueness from characteristic functions, and joint convergence from scalar projections under AC.

[F7]

The bounded-Lipschitz metric characterizes weak convergence on Polish spaces metrizes finite-dimensional weak convergence; Dominated convergence permits integrable domination; Markov's inequality for random variables gives Markov's bound.

Proof

1.1F1F2givenchoose

Choose a deterministic $b_{n} \to \infty$ sufficiently slowly that $b_{n} \sqrt{d} e_{n} \to 0$ and $b_{n} e_{n} d^{2 r + 1} / \sqrt{n} \to 0$ in the Gaussian case. Such a choice exists: the first unmultiplied expression tends to zero because $α > 1 / 2$ , and the second is bounded by a logarithmic factor times $d^{2 r - 2 α + 1 / 2} \to 0$ . Set $D_{n} = {∥ θ - θ_{*} ∥_{2} \leq b_{n} e_{n}}$ in the quotient by constants used in [F1]. Then $Π (D_{n} ∣ X^{n}) \to 1$ in probability. On $D_{n}$ , $∥ θ - θ_{*} ∥_{\infty} \leq C b_{n} \sqrt{d} e_{n} = o (1)$ ; hence $∥ f - f_{0} ∥_{\infty} = o (1)$ uniformly, and the densities are bounded above and away from zero. Write $Π_{D}$ for the posterior conditioned on $D_{n}$ .

2.1givenstep 1.1algebra

For $γ = ψ_{l k}$ , $l \leq L_{n}$ , define $f_{s} = f e^{- s γ / \sqrt{n}} / P_{f} e^{- s γ / \sqrt{n}}$ . For $∣ s ∣ \leq \sqrt{λ_{l}}$ , Taylor expansion of the log moment generating function gives $s \sqrt{n} (P_{f} - P_{n}) γ = ℓ_{n} (f_{s}) - ℓ_{n} (f) + (s^{2} / 2) {Var}_{f} γ + R$ , where $∣ R ∣ \leq C ∣ s ∣^{3} ∥ γ ∥_{\infty} / \sqrt{n}$ . To justify the uniform remainder, differentiate $a \mapsto \log P_{f} e^{a γ}$ three times; its third derivative is the third centered moment under the tilted density, bounded by $C ∥ γ ∥_{\infty} P_{f} γ^{2}$ when $∣ a ∣ ∥ γ ∥_{\infty} = o (1)$ . On $D_{n}$ , $P_{f} γ^{2} \leq C$ by orthonormality and the density bound, while $\sqrt{L_{n}} 2^{L_{n} / 2} / \sqrt{n} \to 0$ . Thus $∣ R ∣ \leq C s^{2}$ and ${Var}_{f} γ \leq C$ . Here $ℓ_{n}$ is the sum log likelihood.

3.1F1step 1.1step 2.1algebra

The map $f \mapsto f_{s}$ is the quotient-coordinate translation $θ \mapsto θ - s (γ - \int γ) / \sqrt{n}$ . On high wavelets it shifts one coordinate; low scaling directions involve only a fixed finite-dimensional block. In S1 the absolute log prior density ratio is at most $C ∣ s ∣ / (\sqrt{n} σ_{l}) \leq C ∣ s ∣$ , since $2^{L_{n} (α + 1 / 2)} \leq \sqrt{n}$ . In S2 it is at most $C [∣ s ∣ (∣ g_{0, l k} ∣ + b_{n} e_{n}) / (\sqrt{n} σ_{l}^{2}) + s^{2} / (n σ_{l}^{2})]$ , with a bounded fixed low-block version. Uniformly in high $l \leq L_{n}$ , its coefficients are bounded: $∣ g_{0, l k} ∣ \leq C 2^{- l (α + 1 / 2)}$ , $b_{n} e_{n} d^{2 r + 1} / \sqrt{n} = o (1)$ , and $d^{2 r + 1} / n = o (1)$ . Changing variables in the numerator likelihood integral and discarding its transformed $D_{n}$ indicator therefore bounds that integral by $e^{C ∣ s ∣ + C s^{2}}$ times the full posterior normalizer. Step 2.1 yields $E_{Π_{D}} e^{s \sqrt{n} (P_{f} - P_{n}) ψ_{l k}} \leq C e^{C s^{2} + C ∣ s ∣} / Π (D_{n} ∣ X^{n})$ . No translation invariance of $D_{n}$ is used.

4.1F2F7step 3.1algebra

Put $Z_{l k} = \sqrt{n} ⟨ f - P_{n}, ψ_{l k} ⟩$ for $l \leq L_{n}$ and $p_{n} = Π (D_{n} ∣ X^{n})$ . Exponential Markov with $s = \pm \sqrt{λ_{l}}$ gives $Π_{D} {∣ Z_{l k} ∣ > u \sqrt{λ_{l}}} \leq C p_{n}^{- 1} e^{- λ_{l} (u - C)}$ . Union over the at most $C 2^{l}$ locations shows $E_{Π_{D}} \max_{k} ∣ Z_{l k} ∣ \leq C p_{n}^{- 1} \sqrt{λ_{l}}$ . Union also over levels and integrate in $u$ above a sufficiently large constant to obtain $E_{Π_{D}} \sup_{l \leq L_{n}, k} ∣ Z_{l k} ∣ / \sqrt{λ_{l}} \leq C / p_{n}$ : the relevant integral is bounded by $C p_{n}^{- 1} \sum_{l \geq 1} 2^{l} \int_{M}^{\infty} e^{- l (u - C)} d u$ , plus a finite low-level term. Also the wavelet level bound and a geometric sum give $E_{Π_{D}} ∥ P_{L_{n}} (f - P_{n}) ∥_{\infty} \leq C p_{n}^{- 1} \sqrt{d λ_{L_{n}} / n}$ .

4.2F1step 1.1step 2.1step 3.1algebra

Fix a finite linear combination $γ$ of wavelets and a fixed real $s$ . Repeat step 2.1 with this bounded $γ$ . Its remainder is now $o (1)$ , and ${Var}_{f} γ \to {Var}_{0} γ$ uniformly on $D_{n}$ . The prior translation involves finitely many fixed coordinates, and its density ratio tends to one uniformly on $D_{n}$ : use the global S1 log-Lipschitz bound or the fixed Gaussian precision block and bounded quotient coordinates in S2. Let $A_{s}$ be the translated image of $D_{n}$ . For large $n$ , $A_{s}$ contains $D_{n}^{-} : = {∥ θ - θ_{*} ∥_{2} \leq b_{n} e_{n} / 2}$ because the translation has norm $O (n^{- 1 / 2}) = o (b_{n} e_{n})$ . Both $Π (D_{n}^{-} ∣ X^{n})$ and $Π (D_{n} ∣ X^{n})$ tend to one by [F1]. Bounding the translated integral above by the full normalizer and below by its integral on $D_{n}^{-}$ therefore gives $E_{Π_{D}} e^{s \sqrt{n} (P_{f} - P_{n}) γ} \to e^{s^{2} {Var}_{0} (γ) / 2}$ in probability.

5.1F2F5F7step 4.1algebra

Each empirical wavelet coordinate has variance at most $D_{0} / n$ and summands bounded by $C 2^{l / 2}$ . Bernstein [F5] gives $\Pr {∣ (P_{n} - P_{0}) ψ_{l k} ∣ > C [\sqrt{t / n} + 2^{l / 2} t / n]} \leq 2 e^{- t}$ . Choose $t = C^{'} (l + 1) + u$ , union over $k$ , and integrate over $u \geq 0$ to get $E \max_{k} ∣ (P_{n} - P_{0}) ψ_{l k} ∣ \leq C [\sqrt{λ_{l} / n} + 2^{l / 2} λ_{l} / n]$ . Summing against $2^{l / 2}$ yields $∥ P_{L_{n}} (P_{n} - P_{0}) ∥_{\infty} = O_{P} (\sqrt{d λ_{L_{n}} / n} + d λ_{L_{n}} / n) = O_{P} (\sqrt{d λ_{L_{n}} / n})$ , since $d λ_{L_{n}} / n \to 0$ . Moreover $∥ P_{L_{n}} f_{0} - f_{0} ∥_{\infty} \leq C d^{- α} \leq C \sqrt{d λ_{L_{n}} / n}$ for both cutoffs. Consequently the posterior projected density error is tight at rate $a_{n} = \sqrt{d λ_{L_{n}} / n}$ .

5.2F3F6F7step 4.2algebra

Here is the needed finite-dimensional Laplace continuity argument. Along any subsequence, extract a further subsequence on which the preceding transform convergences hold almost surely for every rational $s$ and every rational vector of finite-coordinate coefficients, using countability and convergence in probability. The transforms at the positive and negative coordinate unit vectors give tightness of these deterministic finite-dimensional laws by exponential Markov and a finite union bound. [F6] gives a weakly convergent further subsequence. For a fixed rational direction and rational $s$ , its exponential is uniformly integrable: its second moment is bounded by the transform at $2 s$ , and Cauchy–Schwarz and Markov bound the expectation above any large cutoff. Truncation thus passes its transform to the weak limit. In each coordinate direction and then any rational linear direction, the limit has finite exponential moments of every rational order, and hence of every real order by bounding $e^{t ∣ x ∣}$ by $e^{q x} + e^{- q x}$ for rational $q > ∣ t ∣$ . Its moment generating function is analytic: on every bounded complex disk its power series can be integrated term by term under $e^{q ∣ x ∣}$ . Equality with the Gaussian transform on the rationals therefore gives equality of all moments by the real power series at zero. Integrating the characteristic-function power series under the same exponential bound gives the Gaussian characteristic function. [F6] identifies every rational linear projection with the corresponding centered normal. Continuity of characteristic functions in their vector argument extends the identification to real vectors; [F6] then identifies the finite-dimensional law with the Gaussian projection of [F3]. Every weakly convergent further subsequence has that limit, so tightness implies weak convergence of the full deterministic subsequence, and [F7] gives bounded-Lipschitz convergence. The initial subsequence argument proves convergence in probability of the original random projected laws.

5.3F7givenstep 4.1algebra

For the coefficients through $L_{n}$ , step 4.1 and Markov give posterior tightness of $\sup_{l \leq L_{n}, k} ∣ Z_{l k} ∣ / \sqrt{λ_{l}}$ . Therefore for $J < L_{n}$ , the $M (w)$ norm of the part with $J < l \leq L_{n}$ is at most this variable times $\sup_{l > J} \sqrt{λ_{l}} / w_{l}$ , which tends to zero as $J \to \infty$ by admissibility. This proves the required uniform projected posterior tail bound.

6.1F2step 1.1step 5.1algebra

This projected rate controls the full supremum error. Put $h = θ - θ_{*}$ , $f_{*} = e^{θ_{*}} / \int e^{θ_{*}}$ , and $v = h - P_{f_{*}} h \in V_{L_{n}}$ . Since $\int h = 0$ , $∥ h ∥_{\infty} \leq 2 ∥ v ∥_{\infty}$ . Normalized exponential expansion gives $f - f_{*} = f_{*} v + R$ with $∥ R ∥_{\infty} \leq C ∥ h ∥_{\infty}^{2}$ on $D_{n}$ . Also $∥ f_{*} - f_{0} ∥_{\infty} \leq C d^{- α}$ . For any fixed $0 < β < \min (α, 1)$ , $f_{0}$ is $β$ -Hölder, so [F2] gives $∥ P_{L_{n}} (f_{0} v) - f_{0} v ∥_{\infty} \leq C d^{- β} ∥ v ∥_{\infty}$ . The lower bound $f_{0} \geq ρ_{0}$ and uniform boundedness of $P_{L_{n}}$ imply $c ∥ h ∥_{\infty} \leq ∥ P_{L_{n}} (f - f_{0}) ∥_{\infty} + C d^{- β} ∥ h ∥_{\infty} + C ∥ h ∥_{\infty}^{2} + C d^{- α}$ . Step 1.1 makes the two terms proportional to $∥ h ∥_{\infty}$ absorbable. It follows that $∥ f - f_{0} ∥_{\infty} \leq C [∥ P_{L_{n}} (f - f_{0}) ∥_{\infty} + d^{- α}]$ uniformly on $D_{n}$ . Thus the full posterior supremum error is tight at rate $a_{n}$ .

7.1F2givenstep 6.1algebra

Choose $M_{n} \to \infty$ so slowly that $M_{n} \sqrt{λ_{L_{n}}} / w_{L_{n}} \to 0$ . Step 6.1 makes the posterior probability of $∥ f - f_{0} ∥_{\infty} > M_{n} a_{n}$ tend to zero. On its complement, the $L^{1}$ wavelet bound and monotonicity of $w$ give $\sqrt{n} ∥ P_{> L_{n}} (f - f_{0}) ∥_{M (w)} \leq C \sqrt{n} 2^{- L_{n} / 2} w_{L_{n}}^{- 1} M_{n} a_{n} \leq C M_{n} \sqrt{λ_{L_{n}}} / w_{L_{n}} \to 0$ . The truth coefficients satisfy $\sqrt{n} ∥ P_{> L_{n}} f_{0} ∥_{M (w)} \leq C \sqrt{n} d^{- α - 1 / 2} / w_{L_{n}} \leq C \sqrt{λ_{L_{n}}} / w_{L_{n}} \to 0$ for either cutoff. Hence the entire density tail above $L_{n}$ is negligible in posterior probability.

8.1F3F4step 1.1step 5.2step 5.3step 7.1algebra∎

Each density is a finite exponential of bounded continuous functions, hence belongs to $L^{2}$ ; its coefficients, and those of $P_{n} (L_{n})$ , belong to $M_{0} (w)$ . Removing $D_{n}$ changes posterior integrals of functions bounded by one by at most $2 [1 - Π (D_{n} ∣ X^{n})] = o_{P} (1)$ . Step 5.2 supplies fixed finite-projection convergence to [F3], and steps 5.3, 7.1 supply the posterior-tail condition. Apply [F4] to conclude the asserted bounded-Lipschitz convergence in $M_{0} (w)$ .

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

CDF and multiscale credible-band calibration from weak BvM limits

Statement

Let $P_{0}$ have a density $f_{0}$ on $[0, 1]$ bounded above and bounded away from zero, and put $F_{0} (t) = \int_{0}^{t} f_{0} (x) d x$ . Use the wavelet basis and $M_{0} (w)$ of Little-o multiscale coefficient space M0(w), and assume the wavelet conditions and summability condition $\sum_{l} w_{l} 2^{- l / 2} < \infty$ of The multiscale primitive map is continuous into C[0,1]. Let $T_{n}$ be a measurable $L^{2}$ -valued center, let $Π_{n}$ be a posterior kernel supported on $L^{2}$ densities, and define $Q_{n}$ to be the conditional law on $M_{0} (w)$ of $\sqrt{n} (f - T_{n})$ under $f \sim Π_{n} (\cdot ∣ X^{n})$ .

Assume under $P_{0}^{n}$ that (i) $Q_{n}$ converges in bounded-Lipschitz distance in probability to the centered Gaussian bridge coefficient law $G_{P_{0}}$ in $M_{0} (w)$ , whose finite coefficient covariances are $P_{0} [(h - P_{0} h) (g - P_{0} g)]$ , and (ii) $\sqrt{n} (T_{n} - f_{0}) \Rightarrow G_{P_{0}}$ in $M_{0} (w)$ . For $u = 1 - α$ , let $q_{M}$ and $q_{C}$ be the lower $u$ -quantiles of $∥ G_{P_{0}} ∥_{M (w)}$ and $∥ J G_{P_{0}} ∥_{\infty}$ . Assume both CDFs are continuous on $R$ and strictly increasing on open neighborhoods of their respective quantiles, and that $q_{M}, q_{C} > 0$ . These local quantile conditions allow the CDFs to be constant on the negative half-line. In the projected empirical-center case $T_{n} = P_{n} (L_{n})$ , Projected empirical bridge CLT in the little-o multiscale space supplies premise (ii) under its stated smoothness and cutoff conditions. For the source-specific type-S and type-H priors, the posterior BvM premise (i) is established in Type-S density posteriors satisfy the weak multiscale Bernstein–von Mises theorem and Type-H histogram posterior weak Bernstein–von Mises theorem, respectively.

For a posterior draw $f$ , write $F_{f} = J f$ and $F_{T_{n}} = J T_{n}$ , where $J$ is the primitive map of The multiscale primitive map is continuous into C[0,1]. Then the posterior law of $\sqrt{n} (F_{f} - F_{T_{n}})$ converges in bounded-Lipschitz distance on $C ([0, 1])$ to the $P_{0}$ -Brownian bridge $B_{P_{0}} (t)$ , with covariance $F_{0} (s \land t) - F_{0} (s) F_{0} (t)$ .

For $0 < α < 1$ , let $R_{n}^{M}$ and $R_{n}^{C}$ be the posterior lower $1 - α$ quantiles of $∥ \sqrt{n} (f - T_{n}) ∥_{M (w)}$ and $∥ \sqrt{n} (F_{f} - F_{T_{n}}) ∥_{\infty}$ , respectively. The corresponding posterior balls $C_{n}^{M} = {f : ∥ f - T_{n} ∥_{M (w)} \leq R_{n}^{M} / \sqrt{n}}, C_{n}^{C} = {F : ∥ F - F_{T_{n}} ∥_{\infty} \leq R_{n}^{C} / \sqrt{n}}$ are measurable credible sets of posterior mass at least $1 - α$ . Their frequentist coverage tends to $1 - α$ , and the scaled radii converge to the respective $1 - α$ quantiles of $∥ G_{P_{0}} ∥_{M (w)}$ and $∥ B_{P_{0}} ∥_{\infty}$ .

Facts & Assumptions

Given: The density law, posterior kernel, wavelet basis, space $M_{0} (w)$ , primitive map, two weak-convergence assumptions, continuous strictly increasing limit norm CDFs, and $0 < α < 1$ in the Statement.

[F1]

Little-o multiscale coefficient space M0(w): $M_{0} (w)$ has the weighted supremum norm on wavelet coefficients whose normalized level maxima vanish.

[F2]

The multiscale coefficient space is separable and has vanishing projection tails: every square-summable wavelet coefficient sequence belongs to $M_{0} (w)$ .

[F3]

The multiscale primitive map is continuous into C[0,1]: $J : M_{0} (w) \to C ([0, 1])$ is continuous and agrees with $f \mapsto (t \mapsto \int_{0}^{t} f)$ for $L^{2}$ functions.

[F4]

Continuous mapping theorem: a continuous map carries weak convergence of probability laws to weak convergence of their pushforwards.

[F5]

Posterior quantiles are measurable and bracket their probability level: for a posterior probability kernel on $R$ , its lower $u$ -quantile is measurable and its CDF brackets $u$ .

[F6]

Posterior quantiles stabilize under scalar weak convergence: if the limiting scalar CDF is continuous on $R$ and strictly increasing on a neighborhood of the target quantile, bounded-Lipschitz convergence in probability implies convergence of that posterior quantile.

[F7]

Posterior credible set: a graph-measurable rule whose posterior mass is at least $1 - α$ is a posterior credible set at level $1 - α$ .

[A1]

The posterior laws $Q_{n}$ converge in bounded-Lipschitz distance in probability to the centered Gaussian bridge coefficient law $G_{P_{0}}$ in $M_{0} (w)$ .

[A2]

The sampling centers satisfy $\sqrt{n} (T_{n} - f_{0}) \Rightarrow G_{P_{0}}$ in $M_{0} (w)$ .

[A3]

The CDFs of $∥ G_{P_{0}} ∥_{M (w)}$ and $∥ J G_{P_{0}} ∥_{\infty}$ are continuous on $R$ and strictly increasing on open neighborhoods of their positive $1 - α$ quantiles. Continuity at each lower quantile $q$ implies its CDF value is $1 - α$ .

[F8]

Portmanteau theorem: weak convergence of probability measures gives convergence of probabilities of sets whose boundary has zero limit measure.

[F9]

Projected empirical bridge CLT in the little-o multiscale space: for the projected empirical center and the theorem's coefficient regularity and cutoff assumptions, $\sqrt{n} (P_{n} (L_{n}) - f_{0}) \Rightarrow G_{P_{0}}$ in $M_{0} (w)$ .

[F11]

Logarithmic negative Sobolev spaces for white noise: the wavelets form a complete orthonormal basis of $L^{2} ([0, 1])$ , so finite orthogonal projections converge in $L^{2}$ .

[F12]

Cauchy-Schwarz for random variables: square-integrable random variables satisfy $∣ E [U V] ∣ \leq (E U^{2} E V^{2})^{1 / 2}$ .

Proof

1.1A1F3F11F12givenalgebra

The posterior draw $f$ and center $T_{n}$ are $L^{2}$ -valued, so [F2] embeds $\sqrt{n} (f - T_{n})$ in $M_{0} (w)$ ; [F1] supplies its norm. For an $L$ -Lipschitz map $T$ between metric spaces and any admissible bounded 1-Lipschitz test $h$ on its target, $h \circ T / \max (1, L)$ is an admissible test on its domain. Therefore $β_{B L} (T_{#} ν, T_{#} ρ) \leq \max (1, L) β_{B L} (ν, ρ) .$ By [F3], $J$ is linear and Lipschitz with a finite constant $L_{J}$ . The bound applied pointwise to the data-indexed laws in [A1] gives $β_{B L} (J_{#} Q_{n}, J_{#} G_{P_{0}}) \to 0$ in probability. Moreover, if $ℓ : C ([0, 1]) \to R$ is continuous and linear, then $ℓ \circ J$ is continuous and linear on $M_{0} (w)$ . The centered Gaussian premise on $G_{P_{0}}$ says exactly that this scalar pushforward is centered normal, allowing zero variance. Thus $J_{#} G_{P_{0}}$ is a centered Gaussian Borel law. In particular, for any finite $t_{1}, \dots, t_{m}$ and real $a_{1}, \dots, a_{m}$ , the functional $x \mapsto \sum_{i} a_{i} J x (t_{i})$ is continuous and linear, so the evaluation vector $(J G_{P_{0}} (t_{i}))_{i = 1}^{m}$ is jointly centered Gaussian. This uses only the scalar Gaussian pushforward meaning of the premise, without characteristic-function uniqueness or a Gaussian existence theorem. To identify its covariance, let $p_{J} 1_{[0, s]}$ be the finite wavelet projection of the indicator. The finite Gaussian coefficient covariance gives $Cov (J (P_{J} G_{P_{0}}) (s), J (P_{J} G_{P_{0}}) (t)) = P_{0} [(p_{J} 1_{[0, s]} - P_{0} p_{J} 1_{[0, s]}) (p_{J} 1_{[0, t]} - P_{0} p_{J} 1_{[0, t]})] .$ By [F11], $p_{J} 1_{[0, s]} \to 1_{[0, s]}$ in Lebesgue $L^{2}$ ; since $f_{0} \leq B$ , this is also convergence in $L^{2} (P_{0})$ . The variance of a difference of two finite projected Gaussian evaluations is the $P_{0}$ -variance of the corresponding difference of projected indicators, so it tends to zero; hence those evaluations are Cauchy in $L^{2}$ . They also converge almost surely to $J G_{P_{0}} (s)$ because $P_{J} G_{P_{0}} \to G_{P_{0}}$ in $M_{0} (w)$ and $J$ is continuous. The almost-sure and in-probability limits coincide: for any $ε > 0$ , the probability that they differ by more than $ε$ is bounded by the two probabilities that each sequence differs from its limit by more than $ε / 2$ , both tending to zero. If $U_{J}, V_{J}$ are the projected evaluations and $U, V$ their $L^{2}$ limits, [F12] gives $∣ E [U_{J} V_{J}] - E [U V] ∣ \leq ∥ U_{J} - U ∥_{2} ∥ V_{J} ∥_{2} + ∥ U ∥_{2} ∥ V_{J} - V ∥_{2} ⟶ 0;$ the norms $∥ V_{J} ∥_{2}$ are bounded because $V_{J} \to V$ in $L^{2}$ , and the means converge to zero as well. Thus the finite covariance passes to the limit. The resulting covariance is $P_{0} [(1_{[0, s]} - F_{0} (s)) (1_{[0, t]} - F_{0} (t))] = F_{0} (s \land t) - F_{0} (s) F_{0} (t)$ , the defining covariance of the $P_{0}$ -Brownian bridge.

1.2F5F7givenalgebra

The posterior pushforward under either norm is a probability kernel on $[0, \infty)$ . By [F5], the two lower quantiles are measurable and their posterior CDFs at those radii are at least $1 - α$ . Since the centers and norms are measurable, the membership graphs of $C_{n}^{M}$ and $C_{n}^{C}$ are measurable. Thus both sets are credible by [F7].

2.1A1A2A3F1F6F9step 1.1

The norm maps $x \mapsto ∥ x ∥_{M (w)}$ and $F \mapsto ∥ F ∥_{\infty}$ are 1-Lipschitz by the reverse triangle inequality. Applying the pushforward bound of step 1.1 first to the $M (w)$ norm and then to the supremum norm after $J$ gives bounded-Lipschitz convergence in probability of both posterior norm laws to their corresponding limit norm laws. Their CDFs satisfy [A3], so [F6] gives $R_{n}^{M} \to q_{M}$ and $R_{n}^{C} \to q_{C}$ in probability. For the projected empirical center $T_{n} = P_{n} (L_{n})$ , [F9] verifies [A2] under the smoothness and cutoff assumptions in [F9].

3.1A2A3F3F4F8step 2.1algebra∎

By [A2] and [F4], $\sqrt{n} (T_{n} - f_{0}) \Rightarrow G_{P_{0}}$ and its primitive converges weakly to $B_{P_{0}}$ . The frequentist error uses the negative center difference, but $∥ f_{0} - T_{n} ∥_{M (w)} = ∥ T_{n} - f_{0} ∥_{M (w)}$ and $∥ J (f_{0} - T_{n}) ∥_{\infty} = ∥ J (T_{n} - f_{0}) ∥_{\infty}$ because $J$ is linear. Thus each scaled frequentist error norm converges to the corresponding limit norm in [A3]. On the event that the random radius is within $ε$ of its deterministic limit from step 2.1, coverage is bounded above and below by probabilities for the frequentist norm with fixed thresholds expanded or contracted by $ε$ . [F8] gives convergence of those fixed-threshold probabilities because [A3] makes their boundary probabilities zero. Sending $ε ↓ 0$ and using the CDF value $1 - α$ at its quantile from [A3] yields $1 - α$ for both coverage probabilities. This comparison does not require independence between the data-dependent posterior radii and the frequentist center.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Diameter of undersmoothed multiscale credible bands

Statement

Assume all hypotheses of CDF and multiscale credible-band calibration from weak BvM limits at a fixed credibility level $1 - α$ , $0 < α < 1$ , and use its $M_{0} (w)$ credible ball $C_{n}^{M}$ and scaled radius $R_{n}^{M}$ . Let $γ > 0$ , assume the compactly supported wavelets have uniformly bounded overlap and $∥ ψ_{l k} ∥_{\infty} \leq C 2^{l / 2}$ , and define $∥ f ∥_{C_{ψ}^{γ}} : = \sup_{l, k} 2^{l (γ + 1 / 2)} ∣ ⟨ f, ψ_{l k} ⟩ ∣ .$ Suppose $∥ f_{0} ∥_{C_{ψ}^{γ}} < \infty$ . Choose integers $j_{n}$ with $2^{j_{n}} ≍ (n / \log n)^{1 / (2 γ + 1)}$ and let $u_{n} = w_{j_{n}} / \sqrt{j_{n}} \to \infty$ . Intersect the credible ball with the coefficient smoothness ball: ${\overline{C}}_{n} = C_{n}^{M} \cap {f : ∥ f ∥_{C_{ψ}^{γ}} \leq u_{n}} .$ Then $P_{f_{0}}^{(n)} {f_{0} \in {\overline{C}}_{n}} \to 1 - α$ , and its $L^{\infty}$ diameter satisfies ${diam}_{L^{\infty}} ({\overline{C}}_{n}) = O_{P_{f_{0}}^{(n)}} ⁣ ((n / \log n)^{- γ / (2 γ + 1)} u_{n}) .$

Facts & Assumptions

Given: All hypotheses, the credible ball and radius of CDF and multiscale credible-band calibration from weak BvM limits, the coefficient smoothness norm and wavelet assumptions in the Statement, $γ > 0$ , and $u_{n} \to \infty$ .

[F1]

Little-o multiscale coefficient space M0(w): the multiscale ball bounds each level- $l$ coefficient by $w_{l}$ times its radius.

[F2]

CDF and multiscale credible-band calibration from weak BvM limits: $C_{n}^{M}$ has frequentist coverage tending to $1 - α$ , and its scaled radius $R_{n}^{M}$ converges in probability to a finite quantile, hence is $O_{P} (1)$ .

[L1]

The Statement defines $C_{ψ}^{γ}$ by the displayed coefficient norm $\sup_{l, k} 2^{l (γ + 1 / 2)} ∣ ⟨ f, ψ_{l k} ⟩ ∣$ .

Proof

1.1F2L1given

Since $∥ f_{0} ∥_{C_{ψ}^{γ}} < \infty$ and $u_{n} \to \infty$ , [L1] implies the truth belongs to the coefficient smoothness ball for all sufficiently large $n$ . For those $n$ , ${f_{0} \in {\overline{C}}_{n}} = {f_{0} \in C_{n}^{M}}$ , so the asserted coverage follows from [F2].

1.2F1L1givenalgebra

Let $f, g \in {\overline{C}}_{n}$ . For $l \leq j_{n}$ , [F1] gives $\max_{k} ∣ f_{l k} - g_{l k} ∣ \leq 2 R_{n}^{M} w_{l} / \sqrt{n}$ . For $l > j_{n}$ , [L1] and membership in the coefficient smoothness ball give $\max_{k} ∣ f_{l k} - g_{l k} ∣ \leq 2 u_{n} 2^{- l (γ + 1 / 2)}$ . At each point, only a uniformly bounded number of level- $l$ wavelets are nonzero, each bounded by $C 2^{l / 2}$ . Therefore the low-frequency contribution to $∥ f - g ∥_{\infty}$ is at most $C R_{n}^{M} n^{- 1 / 2} \sum_{l \leq j_{n}} 2^{l / 2} w_{l}$ , and the high-frequency contribution is at most $C u_{n} \sum_{l > j_{n}} 2^{- l γ}$ . These series also give uniformly convergent representatives because $γ > 0$ .

2.1step 1.2algebra

Since $w_{l}$ is nondecreasing, $\sum_{l \leq j_{n}} 2^{l / 2} w_{l} \leq C 2^{j_{n} / 2} w_{j_{n}}$ by summing a geometric series. Also $\sum_{l > j_{n}} 2^{- l γ} \leq C_{γ} 2^{- j_{n} γ}$ . The first bound in step 1.2 is thus at most $C R_{n}^{M} w_{j_{n}} 2^{j_{n} / 2} / \sqrt{n} = C R_{n}^{M} u_{n} \sqrt{j_{n}} 2^{j_{n} / 2} / \sqrt{n}$ .

3.1F2step 1.2step 2.1algebra∎

The relation $2^{j_{n}} ≍ (n / \log n)^{1 / (2 γ + 1)}$ implies both $2^{- j_{n} γ} ≍ (n / \log n)^{- γ / (2 γ + 1)}$ and $\sqrt{j_{n}} 2^{j_{n} / 2} / \sqrt{n} \leq C_{γ} (n / \log n)^{- γ / (2 γ + 1)}$ . Combining the low- and high-frequency bounds in steps 1.2 and 2.1 and using $R_{n}^{M} = O_{P} (1)$ from [F2] yields the stated diameter bound.

CorollaryStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Type-S and Type-H density credible bands have asymptotically exact coverage

Statement

Assume AC. Use either Type-S1, Type-S2 or Type-H with all the truth, basis, prior-scale, Dirichlet-parameter and cutoff conditions of Type-S log-density and type-H histogram prior families. Let $w_{l} \geq 1$ be nondecreasing, $w_{l} / \sqrt{l + 1} \to \infty$ , and $\sum_{l} w_{l} 2^{- l / 2} < \infty$ . In Type S take $T_{n} = P_{n} (L_{n})$ , the empirical wavelet projection through level $L_{n}$ . In Type H take the empirical histogram $T_{n} = h_{n}$ at mesh $2^{- L_{n}}$ , equivalently the Haar projection with scaling coordinate and details $l < L_{n}$ .

For each $0 < η < 1$ , the posterior multiscale and CDF balls of CDF and multiscale credible-band calibration from weak BvM limits, at posterior level $1 - η$ and centered at these $T_{n}$ , have posterior mass at least $1 - η$ and frequentist coverage tending to $1 - η$ . Their scaled radii converge in probability to the positive finite $1 - η$ quantiles of the corresponding Gaussian bridge norms. In particular, writing $F_{h} (t) = \int_{0}^{t} h (x) d x$ , the posterior law of $\sqrt{n} (F_{f} - F_{T_{n}})$ converges in bounded-Lipschitz distance on $C ([0, 1])$ to the $P_{0}$ -Brownian bridge law, in $P_{0}^{n}$ -probability.

If additionally the coefficient smoothness and undersmoothing hypotheses of Diameter of undersmoothed multiscale credible bands hold, its intersection with the stated coefficient smoothness ball has the same limiting coverage and the diameter bound asserted there. No unverified entropy, prior-mass, posterior contraction or quantile-regularity premise is required in this corollary.

Facts & Assumptions

Given: AC, one of the three source-specific prior regimes, the specified empirical center, admissible summable weights, and $0 < η < 1$ .

[A1]

The Axiom of Choice is inherited from the following suppliers.

[F1]

Type-S density posteriors satisfy the weak multiscale Bernstein–von Mises theorem and Type-H histogram posterior weak Bernstein–von Mises theorem prove posterior bounded-Lipschitz convergence about the respective empirical projections, to the Borel Gaussian bridge with covariance $P_{0} [(ψ - P_{0} ψ) (ψ^{'} - P_{0} ψ^{'})]$ .

[F2]

Compact wavelet projections approximate smooth functions and commute asymptotically with multiplication preserves coefficient regularity of $e^{g}$ at every positive order, including integers. Projected empirical bridge CLT in the little-o multiscale space proves the sampling CLT for bounded truths with coefficient smoothness $γ > 1 / 2$ and either cutoff regime.

[F3]

Continuous and strictly increasing norm distributions for density bridges proves that both bridge norm CDFs are continuous and strictly increasing on positive arguments and have unique positive finite interior quantiles, under the given positive bounded truth and primitive-map summability conditions.

[F4]

CDF and multiscale credible-band calibration from weak BvM limits gives measurable credible balls, CDF pushforward convergence, radius limits and exact asymptotic coverage from posterior BvM, the sampling-center CLT and the stated norm-quantile conditions.

[F5]

Diameter of undersmoothed multiscale credible bands gives the additional smoothness-ball coverage and diameter conclusion under its explicit coefficient smoothness and undersmoothing assumptions.

Proof

1.1A1F1F2givenalgebra

In Type S, [F1] proves posterior BvM about $T_{n} = P_{n} (L_{n})$ with the locally proved S1 range $α > 1$ and Gaussian S2 range $α > 1 / 2$ , $0 < r < α - 1 / 4$ . The truth $g_{0} = \log f_{0}$ has $α$ coefficient regularity, so [F2] gives that regularity for $f_{0} = e^{g_{0}}$ as well. The retained basis has the support, overlap and supremum bounds required by the empirical-process theorem in [F2]. Taking $γ = α > 1 / 2$ and the given cutoff therefore proves its sampling-center CLT with the same Gaussian bridge.

1.2F1F2givenalgebra

In Type H, [F1] proves posterior BvM about the empirical histogram $h_{n}$ . Haar wavelets have the same support, overlap and supremum bounds. The ordinary $α$ -Hölder truth, $1 / 2 < α \leq 1$ , has Haar coefficients bounded by $C 2^{- l (α + 1 / 2)}$ : subtract its value at one point of the support in the coefficient integral, use the zero mean, support diameter $2^{- l}$ and $L^{1}$ norm $2^{- l / 2}$ . The histogram projection retains details through $J_{n} = L_{n} - 1$ . Both given cutoff regimes remain true with this $J_{n}$ , since multiplying $2^{L_{n}}$ by $1 / 2$ changes only their comparison constants. Thus [F2] with $γ = α$ gives the sampling CLT about this exact histogram center. The finite scaling coordinate is included and its centered fluctuation is zero.

2.1F1F3F4step 1.1step 1.2given

In all three cases, positivity and boundedness of $f_{0}$ , admissibility, and the displayed summability verify [F3]. Hence every norm-quantile assumption of [F4] holds at $u = 1 - η$ . Steps 1.1, 1.2 verify its posterior and sampling premises, with the same coefficient covariance law. Applying [F4] proves all asserted CDF limits, measurable credibility, radius limits and frequentist coverage, for the exact empirical centers and every $0 < η < 1$ .

3.1F5step 2.1given∎

When the additional hypotheses in the last paragraph of the Statement hold, the multiscale credible ball just obtained is precisely the input to [F5]. Applying that theorem yields its smoothness-intersection coverage and stated diameter rate. These are additional hypotheses only for this diameter conclusion; the coverage and CDF conclusions in step 2.1 required no such extra conditions.

5 · Examples, counterexamples and false statements

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