Some strong limit theorems for the largest entries of sample correlation matrices

Abstract

Let $\{X_{k,i};i≥1,k≥1\}$ be an array of i.i.d. random variables and let $\{p_n;n≥1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from $0$ and $∞$. For $W_n=\max _{1≤i<j≤p_n}|∑_{k=1}^nX_{k,i}X_{k,j}|$ and $L_n=\max _{1≤i<j≤p_n}|\hat{ρ}^{(n)}_{i,j}|$ where $\hat{ρ}^{(n)}_{i,j}$ denotes the Pearson correlation coefficient between $(X_{1,i},…,X_{n,i})'$ and $(X_{1,j},…,X_{n,j})'$, the limit laws (i) $\lim_{n\rightarrow \infty}\frac{W_{n}}{n^{\alpha}}=0$ a.s. $(α>1/2)$, (ii) $\lim _{n→∞}n^{1−α}L_n=0$ a.s. $(1/2<α≤1)$, (iii) $\lim_{n\rightarrow \infty}\frac{W_{n}}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\rightarrow \infty}(\frac{n}{\log n})^{1/2}L_{n}=2$ a.s. are shown to hold under optimal sets of conditions. These results follow from some general theorems proved for arrays of i.i.d. two-dimensional random vectors. The converses of the limit laws (i) and (iii) are also established. The current work was inspired by Jiang’s study of the asymptotic behavior of the largest entries of sample correlation matrices.