ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Sample variance from a multivariate delta method

Example

Assume AC. Let $X_{1}, X_{2}, \dots$ be iid real random variables with $E ∣ X_{1} ∣^{4} < \infty$ . Put $μ_{j} = E [X_{1}^{j}]$ for $1 \leq j \leq 4$ , $σ^{2} = μ_{2} - μ_{1}^{2}$ , ${\overset{ˉ}{X}}_{n} = n^{- 1} \sum_{i} X_{i}$ , $Q_{n} = n^{- 1} \sum_{i} X_{i}^{2}$ , and $S_{n}^{2} = Q_{n} - {\overset{ˉ}{X}}_{n}^{2}$ . Then $\sqrt{n} (S_{n}^{2} - σ^{2}) \Rightarrow N_{1} (0, τ^{2}), τ^{2} = Var ((X_{1} - μ_{1})^{2}) = E [(X_{1} - μ_{1})^{4}] - σ^{4} = μ_{4} - 4 μ_{1} μ_{3} + 8 μ_{1}^{2} μ_{2} - 4 μ_{1}^{4} - μ_{2}^{2} .$ Here $σ^{4} = (σ^{2})^{2}$ and $τ^{2} = Var ((X_{1} - μ_{1})^{2})$ .

Facts & Assumptions

Given: AC, iid real observations, and the finite fourth absolute moment.

[A1]

AC is assumed so the library's multivariate iid central limit theorem and multivariate normal law apply.

[A2]

$E ∣ X_{1} ∣^{4} < \infty$ .

[F1]

Identical distribution and IID families: iid means a common law and mutual independence of the whole family.

[F2]

Measurable coordinatewise functions preserve independence: measurable coordinatewise maps of an independent family remain independent.

[F3]

Multivariate iid central limit theorem: iid finite-dimensional vectors with finite second moment have a centered Gaussian weak limit with their covariance, under AC.

[F4]

Multivariate normal law, including singular covariance: every scalar projection of a centered multivariate normal law is normal with the projected covariance.

[F5]

First- and second-order delta method for random vectors: a first-order expansion of a map at the target transfers a vector weak limit through its derivative.

[F6]

The Axiom of Choice: AC states that every family of nonempty sets has a choice function.

[F7]

Moments, variance, and covariance on a probability space: for square-integrable $W$ , $Var (W) = E [(W - E W)^{2}]$ ; covariance is defined by the corresponding centered product.

Verification

Given: The data, moments, sample variance, and assumptions in the example.

1.1A1A2F1F2F3F6

Let $Y_{i} = (X_{i}, X_{i}^{2})$ . The map $x \mapsto (x, x^{2})$ is Borel, so [F1]–[F2] show that $(Y_{i})$ is iid. By [A2], $E ∥ Y_{1} ∥^{2} = μ_{2} + μ_{4} < \infty$ , with mean $(μ_{1}, μ_{2})$ and covariance $C = (\begin{array}{cc} μ_{2} - μ_{1}^{2} & μ_{3} - μ_{1} μ_{2} \\ μ_{3} - μ_{1} μ_{2} & μ_{4} - μ_{2}^{2} \end{array}) .$ By [A1] and [F3], $\sqrt{n} {({\overset{ˉ}{X}}_{n}, Q_{n}) - (μ_{1}, μ_{2})} \Rightarrow Z \sim N_{2} (0, C)$ .

2.1F5step 1.1givenalgebra

The map $g (x, y) = y - x^{2}$ has $g (μ_{1}, μ_{2}) = σ^{2}$ and derivative $(u, v) \mapsto - 2 μ_{1} u + v$ . Apply [F5] to step 1.1 to get $\sqrt{n} (S_{n}^{2} - σ^{2}) \Rightarrow - 2 μ_{1} Z_{1} + Z_{2}$ .

3.1F4F7step 1.1step 2.1givenalgebra∎

By [F4], the limit is centered normal with variance $(- 2 μ_{1}, 1) C (- 2 μ_{1}, 1)^{T} = 4 μ_{1}^{2} (μ_{2} - μ_{1}^{2}) - 4 μ_{1} (μ_{3} - μ_{1} μ_{2}) + μ_{4} - μ_{2}^{2} = μ_{4} - 4 μ_{1} μ_{3} + 8 μ_{1}^{2} μ_{2} - 4 μ_{1}^{4} - μ_{2}^{2}$ . Expanding $E [(X_{1} - μ_{1})^{4}] - (μ_{2} - μ_{1}^{2})^{2}$ gives the same expression. By [F7], it is $Var ((X_{1} - μ_{1})^{2})$ , so the displayed $τ^{2}$ is exactly the limit variance.

Remarks

The cited Duchi sample-variance example supplies the setup and covariance matrix, but its printed final variance $μ_{4} - μ_{2}^{2}$ omits the nonzero-mean terms. Step 3.1 derives the corrected variance directly; the printed expression agrees with it when $μ_{1} = 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 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

John Duchi, Theory of Statistics, Stats 300B Lecture 2, sample-variance setup and covariance, PDF p. 3 (printed final variance corrected here) (standard reference, not scraped)
Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 16, delta method, PDF pp. 10–12 (standard reference, not scraped)