ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

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An anisotropic population with a balanced deterministic direction

Example

Assume the Axiom of Choice. Let $(ξ_{i j})_{i, j \geq 1}$ be one infinite iid complex array with mean zero, unit second absolute moment and finite fourth moment. Let $N_{n}$ be a positive integer with $n / N_{n} \to c \in (0, \infty)$ . For each $n$ , let $T_{n}$ be diagonal with $⌈ n / 2 ⌉$ diagonal entries equal to $1$ and the remaining entries equal to $2$ , and let $x_{n} = n^{- 1 / 2} (1, \dots, 1)^{T}$ . For $A_{n} = N_{n}^{- 1} T_{n}^{1 / 2} X_{n} X_{n}^{*} T_{n}^{1 / 2}$ , the directional spectral measure in direction $x_{n}$ converges almost surely to the deformed Marchenko–Pastur law $F_{c, H}$ with $H = (δ_{1} + δ_{2}) / 2$ .

Example references

Sample-covariance resolvent deterministic equivalents

Directional spectral measure of a Hermitian matrix

Facts & Assumptions

Given: AC, the nested complex iid entry law, the stated diagonal populations, and the deterministic directions.

[A1]

The Axiom of Choice (The Axiom of Choice).

[F1]

The directional theorem requires convergence of the population directional resolvent to the transform of the limiting population law (Sample-covariance resolvent deterministic equivalents).

[F2]

The directional spectral measure assigns mass $∥ P_{λ} x ∥_{2}^{2}$ to each eigenvalue $λ$ (Directional spectral measure of a Hermitian matrix).

[F3]

The population Stieltjes transform is $m_{H} (w) = \int (t - w)^{- 1} d H (t)$ off the nonnegative real axis (Stieltjes transform of a probability measure on the real line).

Verification

Given: The matrices and directions in the Example.

1.1givenF1F3algebra

The empirical eigenvalue measures $H_{n}$ of $T_{n}$ converge to $H = (δ_{1} + δ_{2}) / 2$ , since the proportions of diagonal entries equal to $1$ and $2$ both tend to $1 / 2$ . For every $w \in C ∖ [0, \infty)$ , direct diagonal multiplication gives $x_{n}^{*} (T_{n} - w I)^{- 1} x_{n} = \frac{⌈ n / 2 ⌉}{n (1 - w)} + \frac{⌊ n / 2 ⌋}{n (2 - w)} ⟶ \frac{1}{2} \frac{1}{1 - w} + \frac{1}{2} \frac{1}{2 - w} = m_{H} (w) .$ The vectors have norm one, so all hypotheses of [F1] hold with the transform convention [F3].

2.1givenA1F1F2step 1.1∎

Applying [F1] to the calculation in step 1.1 proves almost-sure weak convergence of the directional spectral measures to the asserted $F_{c, H}$ .

Source notes

Theorem 1 of Bai, Miao, and Pan (full paper PDF).

Theorem 1.1 of Silverstein (full paper PDF).

Depends on

Used by

Nothing in the library uses this result yet.

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Sources

Z. D. Bai, B. Q. Miao, and G. M. Pan, On Asymptotics of Eigenvectors of Large Sample Covariance Matrix (standard reference, not scraped)
J. W. Silverstein, Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices (standard reference, not scraped)