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A Cameron–Martin shift cannot increase centered Gaussian ball mass

Statement

Assume the Axiom of Choice, the Hahn–Banach theorem, and countable choice. Let $B$ be a real separable Banach space, and let $W$ be a centered Borel Gaussian random element with Borel law $μ$ . Let $C : B^{*} \to B$ be its covariance operator, let $E$ be the completion of $B^{*} / \ker C$ with inner product $⟨ [ℓ], [m] ⟩_{E} = m (C ℓ)$ , and let $i : E \to B$ be the canonical injection extending $[ℓ] \mapsto C ℓ$ . For every $h \in E$ and $r > 0$ , $μ ({x \in B : ∥ x - i h ∥_{B} \leq r}) \leq μ ({x \in B : ∥ x ∥_{B} \leq r}) .$

Statement references

a centered Borel Gaussian is Borel measurable and has centered normal continuous-linear projections

the covariance operator and covariance-induced Hilbert space

the canonical Cameron–Martin injection is continuous and injective

Facts & Assumptions

Given: AC, HB, countable choice, a real separable Banach space $B$ , a centered Borel Gaussian random element $W$ , its covariance operator $C$ , and its covariance-induced Hilbert space $E$ with canonical injection $i$ .

[A1]

AC is assumed by the finite-dimensional Gaussian-centering result used below. The Axiom of Choice

[A2]

HB and countable choice supply the countable norming family used to recover the Banach norm from scalar coordinates. The real dominated-extension principle as an additional hypothesis over ZF, The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

$W$ is Borel measurable, each $ℓ (W)$ is centered normal, and every finite vector of continuous-linear coordinates is a possibly degenerate multivariate normal vector. Centered Borel Gaussian random element on a separable Banach space, Multivariate normal law, including singular covariance

[F2]

$E$ is the completion of $B^{*} / \ker C$ , the map $i$ extends $[ℓ] \mapsto C ℓ$ , and $ℓ (i h) = ⟨ h, [ℓ] ⟩_{E}$ . Covariance operator and reproducing-kernel Hilbert space of a Banach Gaussian, The Banach Gaussian covariance and RKHS construction is well-defined

[F3]

There are $ℓ_{j} \in B^{*}$ with $∥ x ∥_{B} = \sup_{j} ∣ ℓ_{j} (x) ∣$ for every $x \in B$ . A separable Banach space has a countable norming family

[F4]

For a nondegenerate finite-dimensional centered Gaussian vector, every bounded symmetric loss with convex sublevel sets has minimum expectation at the center. Applying this to the indicator of the complement of a closed symmetric convex set gives the Gaussian shift inequality for that set. A centered Gaussian minimizes bounded symmetric quasi-convex loss

[F5]

For a finite measure, probabilities of a decreasing sequence of measurable sets converge to the probability of their intersection. Continuity from above when one set has finite measure

Proof

technique · compare finite collections of norming coordinates by finite-dimensional Gaussian centering, then pass to the norm ball by continuity from above

1.1A1A2F1F2F3givenchoose

Choose the norming family from [F3]. Fix $r > 0$ , $h \in E$ , and $n \geq 1$ . Write $Y_{n} = (ℓ_{1} (W), \dots, ℓ_{n} (W))$ , let $Σ_{n}$ be its covariance matrix, and put $a_{n} = (ℓ_{1} (i h), \dots, ℓ_{n} (i h))$ . The vector $Y_{n}$ is centered Gaussian by [F1]. If $c \in \ker Σ_{n}$ , then $\sum_{j \leq n} c_{j} ℓ_{j} (W)$ has variance zero and hence is zero almost surely. For $ℓ_{c} = \sum_{j \leq n} c_{j} ℓ_{j}$ , this gives $C ℓ_{c} = 0$ and $[ℓ_{c}] = 0$ in $E$ ; therefore [F2] gives $c \cdot a_{n} = ℓ_{c} (i h) = ⟨ h, [ℓ_{c}] ⟩_{E} = 0$ . Thus $a_{n} ⊥ \ker Σ_{n}$ , so $a_{n} \in S_{n} : = ran (Σ_{n})$ , the support subspace of $Y_{n}$ .

2.1A1F4step 1.1givenalgebra

In the finite-dimensional space $S_{n}$ , the restriction of $Y_{n}$ is a nondegenerate Gaussian vector. Let $Q_{n} = [- r, r]^{n}$ and $D_{n} = Q_{n} \cap S_{n}$ , a closed symmetric convex set. Since $a_{n} \in S_{n}$ , the event $Y_{n} - a_{n} \in Q_{n}$ is the translate by $a_{n}$ of $D_{n}$ inside $S_{n}$ . An invertible linear change of coordinates turns the centered law on $S_{n}$ into a nondegenerate Gaussian law and preserves symmetry and convexity of $D_{n}$ . Apply [F4] to the bounded loss $1_{D_{n}^{c}}$ in those coordinates: its sublevel sets are $D_{n}$ and the whole space, so it is symmetric and quasi-convex. This yields $P (Y_{n} - a_{n} \in Q_{n}) \leq P (Y_{n} \in Q_{n}) .$ If $S_{n} = {0}$ , then $Y_{n} = a_{n} = 0$ almost surely and the same inequality is equality.

3.1F3F5step 2.1givenalgebra∎

Define $A_{n} (h) = {\max_{j \leq n} ∣ ℓ_{j} (W - i h) ∣ \leq r}$ . Step 2.1 says $P (A_{n} (h)) \leq P (A_{n} (0))$ . The sets $A_{n} (h)$ decrease with $n$ , and [F3] gives $⋂_{n} A_{n} (h) = {∥ W - i h ∥_{B} \leq r}$ ; likewise $⋂_{n} A_{n} (0) = {∥ W ∥_{B} \leq r}$ . Apply [F5] to both decreasing sequences and pass to their limits in the finite-dimensional inequalities. This proves the stated ball-mass bound.

Depends on

Used by

Small-ball ratios for Cameron–Martin translates of a Banach Gaussian Lemma

Dependency tree · two levels

53 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Giné and R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models (standard reference, not scraped)
M. Dashti, K. J. H. Law, A. M. Stuart, and J. Voss, MAP Estimators and Their Consistency in Bayesian Nonparametric Inverse Problems (standard reference, not scraped)