A Strong Limit Theorem of the Largest Entries of a Sample Correlation Matrices under a Strong Mixing Assumption

Abstract

We are interested in an n by p matrix Xn where the n rows are strictly stationary α-mixing random vectors and each of the p columns is an independent and identically distributed random vector; p=pn goes to infinity as n→∞, satisfiying 0<c1≤pn/nτ≤c2<∞, where τ>0, c2≥c1>0. We obtain a logarithmic law of Ln=max1≤i<j≤pn|ρij| using the Chen–Stein Poisson approximation method, where ρij denotes the sample correlation coefficient between the ith column and the jth column of Xn.