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Marchenko–Pastur law and edges below 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 $p_{n} / n \to 1 / 4$ . For $S_{n} = X_{n}^{T} X_{n} / n$ , the limiting empirical spectral distribution has density $ρ_{1 / 4} (x) = {\begin{cases} \frac{2 \sqrt{(9 / 4 - x) (x - 1 / 4)}}{π x}, & 1 / 4 < x < 9 / 4, \\ 0, & otherwise, \end{cases}$ and no atom at zero; the largest eigenvalue tends almost surely to $9 / 4$ , the smallest positive eigenvalue tends almost surely to $1 / 4$ , and $S_{n}$ has no zero eigenvalues eventually almost surely.

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 $γ = 1 / 4$ .

[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-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 $\sqrt{γ} = 1 / 2$ , the endpoints are $a = (1 - 1 / 2)^{2} = 1 / 4$ and $b = (1 + 1 / 2)^{2} = 9 / 4$ , while the zero atom is $(1 - 4)_{+} = 0$ . Substituting in [F1] gives the displayed density on $(1 / 4, 9 / 4)$ and zero outside.

2.1A1L1L2L3step 1.1algebra∎

The nested array and AC satisfy [L1]–[L3]. The MP weak law [L1] gives the limiting density computed in step 1.1. Independently, [L2] gives largest edge $9 / 4$ , while [L3] gives smallest positive edge $1 / 4$ and eventual full column rank, hence no zero eigenvalues. The intersection of these probability-one events proves all 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)