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Almost-sure limit of the largest sample-covariance eigenvalue

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 $p_{n} / n \to γ \in (0, \infty)$ . For $S_{n} = X_{n}^{T} X_{n} / n$ , the largest eigenvalue satisfies $λ_{\max} (S_{n}) ⟶ (1 + \sqrt{γ})^{2} almost surely .$

Statement references

Yin, Bai, and Krishnaiah, Theorem 3.1 and §4, printed pp. 517–520 (complete paper PDF).

Bai and Silverstein, Theorem 5.10, printed pp. 105–112 (complete book PDF).

Facts & Assumptions

[F1]

Finite-fourth-moment upper bound for the largest sample-covariance eigenvalue gives $lim sup λ_{\max} (S_{n}) \leq (1 + \sqrt{γ})^{2}$ .

[F2]

Independent lower bound for the largest sample-covariance eigenvalue gives the independent reverse inequality for the liminf.

Proof

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

1.1givenF1F2

The two suppliers apply to exactly the same upper-left blocks, law, normalization and aspect ratio as in the Statement. Intersect their probability-one events to obtain the upper bound in F1 and lower bound in F2 simultaneously.

2.1step 1.1algebra∎

On this event the liminf and limsup of $λ_{\max} (S_{n})$ coincide with $(1 + \sqrt{γ})^{2}$ . Thus the sequence converges to that value almost surely.

Depends on

Used by

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Sources

Yin, Bai, and Krishnaiah, On the Limit of the Largest Eigenvalue of the Large Dimensional Sample Covariance Matrix, Theorem 3.1 (standard reference, not scraped)
Bai and Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed., Chapter 5, §5.2 (standard reference, not scraped)