LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Mean and covariance determine Gaussian finite-dimensional laws

Statement

Assume the Axiom of Choice. Two real Gaussian processes on the same index set that have the same mean function and covariance function have identical finite-dimensional distributions.

Facts & Assumptions

Given: AC and Gaussian processes $X, Y$ satisfying the two equalities in the Statement.

[F1]

Every finite evaluation vector of a Gaussian process has a possibly singular multivariate normal law. Gaussian process

[F2]

Assume AC. The characteristic function of $N_{n} (m, Σ)$ is $u \mapsto \exp (i u \cdot m - u^{T} Σ u / 2)$ , and it uniquely determines the law, including when $Σ$ is singular. Characteristic function of a multivariate normal law

Proof

technique · direct

1.1

Fix $n \geq 1$ and times $t_{1}, \dots, t_{n} \in I$ . By [F1], the vectors $X^{(n)} = (X_{t_{1}}, \dots, X_{t_{n}}), Y^{(n)} = (Y_{t_{1}}, \dots, Y_{t_{n}})$ are multivariate normal. Their mean vectors agree by the first given identity. Their covariance matrices agree entry by entry by the second identity, even if some times repeat and the common matrix is singular.

givenF1

2.1

Write the common mean vector and covariance matrix as $m$ and $Σ$ . By [F2], both vector characteristic functions equal $u ⟼ \exp (i u \cdot m - u^{T} Σ u / 2) .$ The uniqueness clause of [F2] therefore gives $X^{(n)} \overset{d}{=} Y^{(n)}$ . Since the finite time list was arbitrary, all finite-dimensional distributions agree. The empty-coordinate law, if included as a convention, is the unique probability law on the singleton empty tuple. AC is used exactly through [F1]–[F2], not to choose a version of either process.

step 1.1F1F2∎

Source notes

Sousi, Section 6.1, records that the mean and covariance functions determine a Gaussian process in law. The proof above supplies the complete singular-law argument via the library's multivariate characteristic-function theorem.

Depends on

Used by

Brownian time inversion Theorem
Uniqueness of Wiener measure Theorem

Dependency tree · two levels

10 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

Perla Sousi, Advanced Probability, Section 6.1 (standard reference, not scraped)