LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generated

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 two-simulation covariance in two dimensions is singular

Statement

For any two simulated vectors $z_{1}, z_{2} \in R^{2}$ , put $\overset{ˉ}{z} = (z_{1} + z_{2}) / 2$ and define the Bessel-corrected sample covariance $\hat{Σ} : = \sum_{i = 1}^{2} (z_{i} - \overset{ˉ}{z}) (z_{i} - \overset{ˉ}{z})^{T} .$ Then $\hat{Σ}$ is not positive definite.

Facts & Assumptions

Given: Two vectors $z_{1}, z_{2} \in R^{2}$ and the covariance matrix defined in the Statement.

[F1]

A real symmetric matrix is positive definite when its quadratic form is strictly positive on every nonzero vector. Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal

Proof

technique · direct

1.1givenconstructalgebra

Set $v = (z_{1} - z_{2}) / 2$ . Then $z_{1} - \overset{ˉ}{z} = v$ and $z_{2} - \overset{ˉ}{z} = - v$ , so $\hat{Σ} = 2 v v^{T}$ .

2.1F1step 1.1choosealgebra∎

Write $v = (a, b)$ . If $v \neq 0$ , choose $u = (- b, a)$ ; if $v = 0$ , choose $u = (1, 0)$ . In either case $u \neq 0$ , $u^{T} v = 0$ , and $u^{T} \hat{Σ} u = 2 (u^{T} v)^{2} = 0$ . This contradicts the strict quadratic-form condition in [F1], so $\hat{Σ}$ is not positive definite.

Depends on

Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal

Used by

Dependency tree · two levels

3 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

David T. Frazier et al., Bayesian Inference Using Synthetic Likelihood: Asymptotics and Adjustments, Journal of the American Statistical Association 118(544), 2821–2832 (2023) (standard reference, not scraped)
Victor M.-H. Ong et al., Variational Bayes with Synthetic Likelihood, Statistics and Computing 28(4), 971–988 (2018) (standard reference, not scraped)