LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

Weighted chi-square triangular-array limit

Statement

Assume the Axiom of Choice. For each $n$ , let $G_{n, 1}, \dots, G_{n, m_{n}}$ be independent standard normal random variables, and let $w_{n, j} \geq 0$ satisfy $2 \sum_{j = 1}^{m_{n}} w_{n, j}^{2} = 1$ . If $\max_{1 \leq j \leq m_{n}} w_{n, j} \to 0$ , then $T_{n} = \sum_{j = 1}^{m_{n}} w_{n, j} (G_{n, j}^{2} - 1)$ has mean zero and variance one for every $n$ and converges in distribution to $N (0, 1)$ .

Statement references

Ingster, Sapatinas and Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, equations (4.2)–(4.3)

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.3, Gaussian white-noise signal detection

Facts & Assumptions

Given: AC; for each $n$ , independent standard normal coordinates $G_{n, j}$ ; nonnegative weights with $2 \sum_{j} w_{n, j}^{2} = 1$ and $\max_{j} w_{n, j} \to 0$ .

[A1]

The Axiom of Choice: AC is the hypothesis of the Lindeberg–Feller supplier.

[F1]

A normal variable has finite fourth moment: for $G \sim N (0, 1)$ , $E G^{2} = 1$ and $E G^{4} = 3$ ; hence $V = G^{2} - 1$ has mean $0$ , variance $2$ , and finite second moment.

[F2]

Lindeberg-Feller central limit theorem: sufficiency: an independent centered triangular array with positive row standard deviations and the Lindeberg condition has its normalized row sums converging in distribution to $N (0, 1)$ .

[F3]

Dominated convergence: an integrable dominating function allows limits to pass through the expectation.

Proof

1.1givenF1algebra

By [F1], each $V_{n, j} = G_{n, j}^{2} - 1$ is centered with variance $2$ . Independence within each row gives $E T_{n} = 0$ and $Var (T_{n}) = 2 \sum_{j} w_{n, j}^{2} = 1$ . Thus the row standard deviation is $s_{n} = 1 > 0$ .

2.1givenF1F3step 1.1algebra

Fix $δ > 0$ and put $w_{n}^{*} = \max_{j} w_{n, j}$ . If $w_{n, j} = 0$ , its Lindeberg summand is zero. Otherwise $w_{n, j} ∣ V_{n, j} ∣ > δ$ implies $∣ V_{n, j} ∣ > δ / w_{n}^{*}$ . Therefore $\sum_{j} E ⁣ [w_{n, j}^{2} V_{n, j}^{2} 1_{{w_{n, j} ∣ V_{n, j} ∣ > δ}}] \leq \frac{1}{2} E ⁣ [V^{2} 1_{{∣ V ∣ > δ / w_{n}^{*}}}] .$ Because $w_{n}^{*} \to 0$ and $V^{2}$ is integrable by [F1], the right side tends to zero by [F3]. This is the Lindeberg condition for the array $w_{n, j} V_{n, j}$ .

3.1A1F2step 1.1step 2.1∎

The array in step 2.1 is centered and independent within rows, has finite second moments, and has row standard deviation $1$ . By [F2] and the Lindeberg condition just proved, $T_{n} \Rightarrow N (0, 1)$ .

Depends on

Used by

Sharp Gaussian asymptotics under a vanishing largest weight Corollary

Dependency tree · two levels

36 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

Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, equations (4.2)–(4.3) (standard reference, not scraped)
Evarist Giné and Richard Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.3 (standard reference, not scraped)