Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero

TL;DR

This paper proves that the largest eigenvalue of a normalized sample covariance matrix converges to 1 almost surely when both the dimension and sample size grow infinitely with their ratio approaching zero.

Contribution

It establishes the almost sure convergence of the largest eigenvalue under the regime where the dimension grows faster than the sample size, with the ratio tending to zero.

Findings

Largest eigenvalue converges to 1 almost surely

Convergence holds when p/n approaches zero

Results extend understanding of eigenvalue behavior in high dimensions

Abstract

Let $\mathbf{X}_p=(\mathbf{s}_1,...,\mathbf{s}_n)=(X_{ij})_{p \times n}$ where $X_{ij}$'s are independent and identically distributed (i.i.d.) random variables with $EX_{11}=0,EX_{11}^2=1$ and $EX_{11}^4<\infty$. It is showed that the largest eigenvalue of the random matrix $\mathbf{A}_p=\frac{1}{2\sqrt{np}}(\mathbf{X}_p\mathbf{X}_p^{\prime}-n\mathbf{I}_p)$ tends to 1 almost surely as $p\rightarrow\infty,n\rightarrow\infty$ with $p/n\rightarrow0$.