LemmaStatement: AI-adaptedProof: 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.

Characteristic function of a multivariate normal law

Statement

Assume AC. If $X \sim N_{d} (m, Σ)$ , then $Φ_{X} (t) : = E e^{i t \cdot X} = \exp (i t \cdot m - t^{T} Σ t / 2) (t \in R^{d}) .$ This transform uniquely determines the law, including singular Sigma.

Facts & Assumptions

[F1]

Every linear projection has the specified scalar normal law, and the vector law exists. Multivariate normal law, including singular covariance.

[F2]

A scalar normal has transform exp(ims-sigma^2s^2/2). Characteristic function of a normal law.

[F3]

Under AC scalar laws with equal characteristic functions agree. Uniqueness of a law from its characteristic function.

[F4]

Under AC all linear projection laws determine the Borel vector law. Cramer wold device.

Proof

Given: Assume AC. If $X \sim N_{d} (m, Σ)$ , then $Φ_{X} (t) : = E e^{i t \cdot X} = \exp (i t \cdot m - t^{T} Σ t / 2) (t \in R^{d}) .$ This transform uniquely determines the law, including singular Sigma.

1.1

For fixed t, [F1] makes $t \cdot X$ scalar normal of mean $t \cdot m$ and variance $t^{T} Σ t$ . Evaluate its characteristic function from [F2] at scalar frequency one. This gives the displayed formula. If t=0 both sides are one; if $t^{T} Σ t = 0$ the scalar law is the point mass at $t \cdot m$ and the same formula applies.

F1F2

2.1

Let Y have another Borel probability law with the same displayed vector transform. For every u and scalar s, $φ_{u \cdot Y} (s) = Φ_{Y} (s u) = Φ_{X} (s u) = φ_{u \cdot X} (s)$ . Scalar uniqueness [F3] identifies each pair of projection laws. Then [F4] identifies the vector laws. Thus the construction in [F1] is independent of any realization choices. No determinant or inverse of Sigma is used. AC is inherited from [F1]–[F4]; in dimension zero the single possible law makes the assertion immediate.

step 1.1F1F3F4∎

Depends on

Used by

A degenerate multivariate Gaussian limit Example

Dependency tree · two levels

52 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

Norris, Probability and Measure, Theorem 8.2.1 (standard reference, not scraped)
Aldous and Chewi, Probability Theory notes, Theorem 8.2 (standard reference, not scraped)