ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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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]

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 every finite orthonormal tuple.

[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.

Depends on

Used by

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Sources

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Gaussian Sequence Space Model, equation (1.8), PDF pp. 21–22 (standard reference, not scraped)
Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, §1.2, equations (7)–(8), printed pp. 2005–2006 (standard reference, not scraped)