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Finite-fourth-moment upper bound for 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 $S_{n} = X_{n}^{T} X_{n} / n$ , with $b_{γ} = (1 + \sqrt{γ})^{2}$ , the upper-edge bound is ${lim sup}_{n \to \infty} λ_{\max} (S_{n}) \leq b_{γ} almost surely .$

Statement references

Yin, Bai, and Krishnaiah, Theorem 3.1 and its high-moment proof in §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 covariance spectral confinement gives a sharp norm confinement for the smaller Gram matrix after normalization by the larger rectangular dimension.

[F2]

Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition and The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector identify the largest eigenvalue of either Gram matrix with the squared operator norm; these positive eigenvalues coincide.

Proof

Given: The nested iid array, its upper-left rectangular blocks, and the hypotheses in the Statement.

1.1givenF1F2algebra

Set $N_{n} = \max (n, p_{n})$ , $m_{n} = \min (n, p_{n})$ , $c_{n} = N_{n} / n \to c = \max (1, γ)$ and $y_{n} = m_{n} / N_{n} \to y = \min (γ, γ^{- 1})$ . Orient $X_{n}$ or its transpose as $Z_{n}$ with $m_{n}$ rows and $N_{n}$ columns. By F1, almost surely $lim sup ∥ Z_{n} Z_{n}^{T} / N_{n} - (1 + y) I ∥ \leq 2 \sqrt{y}$ . For each unit vector $v$ , its quadratic form is at most $1 + y$ plus this norm, so $lim sup λ_{\max} (Z_{n} Z_{n}^{T} / N_{n}) \leq (1 + \sqrt{y})^{2}$ .

2.1F2step 1.1algebra∎

By F2, $λ_{\max} (S_{n}) = c_{n} λ_{\max} (Z_{n} Z_{n}^{T} / N_{n})$ . The previous bound gives $lim sup λ_{\max} (S_{n}) \leq c (1 + \sqrt{y})^{2}$ . If $γ \leq 1$ this constant is $(1 + \sqrt{γ})^{2}$ ; if $γ \geq 1$ it is $γ (1 + γ^{- 1 / 2})^{2} = (1 + \sqrt{γ})^{2}$ . The same identity holds at $γ = 1$ , even if the orientation changes infinitely often.

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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)