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Statistical inverse experiments and white noise — Examples and Counterexamples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Two direct white-noise functionals

Example

Take the single-parameter experiment with $H = R^{2}$ , its standard inner product, parameter set ${x}$ , mean signal $K x = (m_{1}, m_{2})$ , and noise level $ϵ > 0$ . Let $e_{1} = (1, 0)$ , $e_{2} = (0, 1)$ , and $g = a e_{1} + b e_{2}$ for $a, b \in R$ . The direct observations $Y_{x} (e_{1})$ and $Y_{x} (e_{2})$ are independent with laws $N (m_{1}, ϵ^{2})$ and $N (m_{2}, ϵ^{2})$ . In $L^{2} (P)$ , $Y_{x} (g) = a Y_{x} (e_{1}) + b Y_{x} (e_{2})$ , with mean $a m_{1} + b m_{2}$ and variance $ϵ^{2} (a^{2} + b^{2})$ ; when $a = b = 0$ , $Y_{x} (g)$ is the constant zero.

Example references

Isonormal white-noise inverse experiment

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.1, PDF pp. 1–2

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §1.2.2, equations (1.6)–(1.8), PDF pp. 21–22

Facts & Assumptions

[F1]

Isonormal Gaussian white-noise inverse experiment: The process is linear into $L^{2} (P)$ , centered, has covariance $E [Z (g) Z (h)] = ⟨ g, h ⟩_{H}$ , and has independent standard-normal coordinates on finite orthonormal tuples.

Given: The two-dimensional experiment, $e_{1}, e_{2}$ , $m_{1}, m_{2}$ , $a, b$ , and $ϵ > 0$ in the Example.

Verification

1.1givenF1

Since $e_{1}, e_{2}$ are orthonormal, [F1] gives independent standard-normal noises $Z (e_{1}), Z (e_{2})$ . Each observation is an invertible affine function of its noise coordinate because $ϵ > 0$ , so it has the same generated sigma-algebra; the affine-image definition of $N$ gives the two asserted laws and preserves independence.

2.1givenF1algebra∎

Linearity of the process gives $Y_{x} (g) = a Y_{x} (e_{1}) + b Y_{x} (e_{2})$ as an equality of its almost-sure equivalence classes. The mean is $a m_{1} + b m_{2}$ , and the covariance identity in [F1] gives variance $ϵ^{2} ∥ a e_{1} + b e_{2} ∥^{2} = ϵ^{2} (a^{2} + b^{2})$ . If $a = b = 0$ , then $g = 0$ and linearity gives $Y_{x} (0) = 0$ .

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Gaussian coefficient sequence

Example

Let $H = ℓ^{2} (N)$ with standard basis $e_{j}$ , take a single parameter with signal $θ = (θ_{j}) \in ℓ^{2} (N)$ , and fix $ϵ > 0$ . The coordinate statistic $Y_{j} = Y (e_{j}) = θ_{j} + ϵ Z (e_{j})$ has independent coordinates with laws $N (θ_{j}, ϵ^{2})$ . Each finite truncation $(Y_{1}, \dots, Y_{N})$ therefore has the product of its $N$ scalar normal laws. The coordinate statistic takes values in $R^{N}$ ; this description does not identify the observation with an $ℓ^{2}$ -valued vector.

Example references

White-noise spectral sequence representation

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Gaussian Sequence Space Model, equation (1.8), PDF pp. 21–22

Ding and Mathé, Minimax rates for statistical inverse problems under general source conditions, §1, equations (1)–(3), PDF pp. 11–12

Facts & Assumptions

[F1]

White-noise spectral sequence representation: In a fixed complete finite or countable orthonormal basis, the observation coordinates are independent with laws $N (⟨ K x, u_{j} ⟩_{H}, ϵ^{2})$ .

Given: The signal $θ \in ℓ^{2} (N)$ , the standard basis of $H$ , and $ϵ > 0$ in the Example.

Verification

1.1givenF1

Apply [F1] with $K x = θ$ and $u_{j} = e_{j}$ to obtain the independent coordinate laws $N (θ_{j}, ϵ^{2})$ for every $j$ .

2.1givenF1algebra∎

Independence of every finite subfamily means that for each $N$ the joint law of $(Y_{1}, \dots, Y_{N})$ is the product of those $N$ marginal normal laws. The sample-space assertion is the coordinate-statistic map into $R^{N}$ , and no step identifies it with an element of $ℓ^{2}$ .

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Finite truncation with its omitted white-noise tail

Example

Let $(u_{j})$ be a fixed complete orthonormal basis, $h = K x$ a mean signal, $g \in H$ a test vector, and $N$ a positive integer. Define $P_{N} g = \sum_{j \leq N} ⟨ g, u_{j} ⟩_{H} u_{j}$ and $r_{N} = g - P_{N} g$ . Then

$Y_{x} (g) = \sum_{j \leq N} ⟨ g, u_{j} ⟩_{H} Y_{x} (u_{j}) + ⟨ h, r_{N} ⟩_{H} + ϵ Z (r_{N})$

in $L^{2} (P)$ . The omitted tail has mean $⟨ h, r_{N} ⟩_{H}$ , variance $ϵ^{2} ∥ r_{N} ∥_{H}^{2}$ , and is independent of the first $N$ coordinate observations. If $r_{N} = 0$ , the omitted tail vanishes.

Example references

White-noise spectral sequence representation

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Gaussian Sequence Space Model, equation (1.8), PDF pp. 21–22

Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, §1.2, equations (7)–(8), printed pp. 2005–2006

Facts & Assumptions

[F1]

[F2]

White-noise spectral sequence representation: For every $g \in H$ , the finite coordinate sums converge to $Y_{x} (g)$ in $L^{2} (P)$ .

Given: The complete orthonormal basis, the test vector $g$ , the mean signal $h = K x$ , the cutoff $N$ , and the isonormal experiment in the Example.

Verification

1.1givenF1F2algebra

Since $g = P_{N} g + r_{N}$ and the process is linear by [F1], $Y_{x} (g) = Y_{x} (P_{N} g) + Y_{x} (r_{N})$ . Applying [F2] to $P_{N} g$ , whose later finite projections equal $P_{N} g$ , gives $Y_{x} (P_{N} g) = \sum_{j \leq N} ⟨ g, u_{j} ⟩_{H} Y_{x} (u_{j})$ ; the definition gives $Y_{x} (r_{N}) = ⟨ h, r_{N} ⟩_{H} + ϵ Z (r_{N})$ . This proves the displayed decomposition.

2.1givenF1algebra∎

The residual $r_{N}$ is orthogonal to $u_{1}, \dots, u_{N}$ . If $r_{N} \neq 0$ , adjoining $r_{N} / ∥ r_{N} ∥_{H}$ to these vectors gives a finite orthonormal tuple, so [F1] makes $Z (r_{N} / ∥ r_{N} ∥_{H})$ independent of the first $N$ coordinate noises; linearity gives $Z (r_{N}) = ∥ r_{N} ∥_{H} Z (r_{N} / ∥ r_{N} ∥_{H})$ as an equality of its almost-sure equivalence classes. Since $ϵ > 0$ , this noise and the tail observation are invertible affine functions of one another, so the tail observation is independent of the first $N$ coordinates. The covariance identity gives its variance $ϵ^{2} ∥ r_{N} ∥_{H}^{2}$ and its mean is $⟨ h, r_{N} ⟩_{H}$ . If $r_{N} = 0$ , both are zero.

Statistical inverse experiments and white noise — Examples and Counterexamples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

Two direct white-noise functionals

Example

Example references

Facts & Assumptions

Verification

Gaussian coefficient sequence

Example

Example references

Facts & Assumptions

Verification

Finite truncation with its omitted white-noise tail

Example

Example references

Facts & Assumptions

Verification

Sources