ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A degenerate multivariate Gaussian limit

Example

Assume AC. Let $Y_{k}$ be centered iid real variables of variance $σ^{2} > 0$ and set $X_{k} = (Y_{k}, Y_{k})$ . Then $n^{- 1 / 2} \sum_{k = 1}^{n} X_{k} \Rightarrow (Z, Z), Z \sim N (0, σ^{2}) .$ The limit covariance is $Σ = σ^{2} (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ , a singular matrix of rank one.

Facts & Assumptions

[F1]

Iid vectors with finite second moments have the Gaussian covariance limit, even if singular. Multivariate iid central limit theorem.

[F2]

A Gaussian law is characterized by its normal projections. Multivariate normal law, including singular covariance.

[F3]

The projection-defined Gaussian law is unique. Characteristic function of a multivariate normal law.

[F4]

Scalar affine images have the stated characteristic functions. Characteristic functions under affine maps and independent sums.

[F5]

Scalar normals have the specified transform and variance. Characteristic function of a normal law.

[F6]

Equal scalar characteristic functions imply equal laws under AC. Uniqueness of a law from its characteristic function.

Verification

Given: Assume AC. Let $Y_{k}$ be centered iid real variables of variance $σ^{2} > 0$ and set $X_{k} = (Y_{k}, Y_{k})$ . Then $n^{- 1 / 2} \sum_{k = 1}^{n} X_{k} \Rightarrow (Z, Z), Z \sim N (0, σ^{2}) .$ The limit covariance is $Σ = σ^{2} (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ , a singular matrix of rank one.

1.1

The vector has mean (0,0), second norm moment $2 E Y_{k}^{2} = 2 σ^{2}$ , and each covariance entry equals $E Y_{k}^{2} = σ^{2}$ . The two columns of Sigma agree and are nonzero, so its rank is one and determinant zero. Its eigenvectors (1,1) and (1,-1) have eigenvalues $2 σ^{2}$ and zero. [F1] therefore gives convergence to $N_{2} (0, Σ)$ .

2.1

For a scalar $Z \sim N (0, σ^{2})$ , every projection of (Z,Z) is $(u_{1} + u_{2}) Z$ . By [F4]–[F6], this has law $N (0, σ^{2} (u_{1} + u_{2})^{2})$ , including a negative coefficient and coefficient zero. Since $u^{T} Σ u = σ^{2} (u_{1} + u_{2})^{2}$ , [F2]–[F3] identify (Z,Z) with the target law. It is supported on the diagonal, and the (1,-1) projection is identically zero both before and after the limit. If the optional case sigma=0 is allowed, all centered Y_k vanish almost surely and the example reduces to the point mass (0,0). AC is inherited from the Gaussian and multivariate CLT suppliers.

step 1.1F2F3F4F5F6∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Norris, Probability and Measure, Section 8.1 (standard reference, not scraped)
Aldous and Chewi, Probability Theory notes, Lecture 8 (standard reference, not scraped)