ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

Structural zeros and the positive edge above aspect one

Example

Assume AC. Let $(ξ_{i j})_{i, j \geq 1}$ be one infinite real iid array with mean zero, variance one and finite fourth moment. Set $X_{n} = (ξ_{i j})_{1 \leq i \leq n, 1 \leq j \leq p_{n}}$ and suppose $p_{n} / n \to 4$ . For $S_{n} = X_{n}^{T} X_{n} / n$ , the limiting empirical spectral distribution has density $ρ_{4} (x) = {\begin{cases} \frac{\sqrt{(9 - x) (x - 1)}}{8 π x}, & 1 < x < 9, \\ 0, & otherwise, \end{cases}$ and an atom of mass $3 / 4$ at zero; almost surely $S_{n}$ has exactly $p_{n} - n$ zero eigenvalues eventually, its largest eigenvalue tends to $9$ , and its smallest positive eigenvalue tends to $1$ .

Example references

Marchenko–Pastur measure with aspect ratio gamma

Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices

Almost-sure limit of the largest sample-covariance eigenvalue

Almost-sure limit and structural-zero count at the lower sample-covariance edge

Facts & Assumptions

Given: AC, the entry-law assumptions in the Example, and $γ = 4$ .

[A1]

The Axiom of Choice (The Axiom of Choice).

[A2]

The Axiom of Choice implies countable choice (AC implies DC implies countable choice).

[F1]

Under countable choice, the density, endpoints, and atom are defined by the Marchenko–Pastur formula (Marchenko–Pastur measure with aspect ratio gamma).

[L1]

The MP weak law is stated in Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices.

[L2]

The largest-edge conclusion is stated in Almost-sure limit of the largest sample-covariance eigenvalue.

[L3]

The smallest-positive-edge and eventual-full-row-rank conclusions are stated in Almost-sure limit and structural-zero count at the lower sample-covariance edge.

Verification

Given: The hypotheses and notation in the Example.

1.1givenA2F1algebra

Since $\sqrt{γ} = 2$ , the endpoints are $a = (1 - 2)^{2} = 1$ and $b = (1 + 2)^{2} = 9$ , while the zero atom is $(1 - 1 / 4)_{+} = 3 / 4$ . Substituting in [F1] gives the displayed density on $(1, 9)$ and zero outside, plus the atom $3 / 4$ .

2.1A1L1L2L3step 1.1algebra∎

The nested array and AC satisfy [L1]–[L3]. The MP weak law [L1] gives the limiting density and atom computed in step 1.1. Independently, [L2] gives largest edge $9$ ; [L3] gives smallest positive edge $1$ and eventual full row rank, hence exactly $p_{n} - n$ zero eigenvalues. Intersecting these probability-one events proves the assertions.

Depends on

Used by

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Sources

Bai and Silverstein, Spectral Analysis of Large Dimensional Random Matrices, Chapters 3 and 5 (standard reference, not scraped)