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

Statement

Let $(ξ_{i j})_{i, j \geq 1}$ be one infinite iid array of real random variables with mean zero, variance one and finite fourth moment. For each positive integer $n$ , let $p_{n}$ be a positive integer and let $X_{n} = (ξ_{i j})_{1 \leq i \leq n, 1 \leq j \leq p_{n}}$ and suppose $p_{n} / n \to γ \in (0, \infty)$ . Set $S_{n} = X_{n}^{T} X_{n} / n$ and let $λ_{\min}^{} (S_{n})$ denote its smallest positive eigenvalue whenever one exists. If $γ < 1$ , then almost surely $X_{n}$ has full column rank for all sufficiently large $n$ , so $S_{n}$ has no zero eigenvalues eventually and $λ_{\min} (S_{n}) \to (1 - \sqrt{γ})^{2}$ . If $γ > 1$ , then almost surely $X_{n}$ has full row rank for all sufficiently large $n$ , so $S_{n}$ has exactly $p_{n} - n$ zero eigenvalues and $λ_{\min}^{} (S_{n}) \to (\sqrt{γ} - 1)^{2}$ . If $γ = 1$ , then almost surely $λ_{\min}^{} (S_{n}) \to 0$ .

Statement references

Bai and Yin, Theorem 1, official publication record; the full paper was not retrievable (publication record).

Bai and Silverstein, Theorem 5.11, printed p. 106 (complete book PDF).

Facts & Assumptions

[F1]

Bai–Yin lower singular-value limits and full rank away from aspect one supplies the lower singular-value limits, eventual full rectangular rank when $γ \neq 1$ , and the eventually defined closing positive edge when $γ = 1$ .

[F2]

Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition and The rank of a linear map is the number of its nonzero singular values identify positive covariance eigenvalues with squared positive singular values divided by $n$ and their number with the rank. The zero multiplicity is therefore $p_{n} - rank X_{n}$ by Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis.

Proof

Given: The nested iid array and hypotheses in the Statement.

1.1givenF1F2algebra

When $γ < 1$ , F1 gives full column rank eventually. Thus F2 gives no covariance zero eigenvalues eventually, and the least eigenvalue is $s_{p_{n}} (X_{n})^{2} / n \to (1 - \sqrt{γ})^{2}$ .

2.1F1F2algebrastep 1.1

When $γ > 1$ , F1 gives full row rank eventually. Then F2 gives exactly $p_{n} - n$ zero eigenvalues and identifies the smallest positive eigenvalue with $s_{n} (X_{n})^{2} / n \to (\sqrt{γ} - 1)^{2}$ .

3.1F1F2algebrastep 2.1∎

When $γ = 1$ , F1 gives an eventually defined smallest positive singular value whose ratio to $\sqrt{n}$ tends to zero. F2 squares this ratio to obtain $λ_{\min}^{} (S_{n}) \to 0$ . Intersecting the relevant probability-one supplier events proves every asserted almost-sure conclusion.

Depends on

Used by

Dependency tree · two levels

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Sources

Bai and Yin, Limit of the Smallest Eigenvalue of a Large-Dimensional Sample Covariance Matrix, official publication record (standard reference, not scraped)
Bai and Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed., Theorem 5.11 (standard reference, not scraped)