A strong limit theorem of the largest entries of sample correlation matrices under a φ-mixing assumption

Abstract

Let {Xk,i ; k ≥ 1, i ≥ 1} be an array of random variables, {Xk ; k ≥ 1} be a strictly stationary φ-mixing sequence, where Xk = (Xk,1 , Xk,2 , · · · , Xk,p ). Let {pn ; n ≥ 1} be a sequence of positive integers suchthat0<c1 ≤pn/nτ ≤c2 <∞,whereτ>0,c2 ≥c1 >0. Inthispaper,weobtainastronglimit theorem of Ln = max1≤i< j≤pn |ρi j |, where ρi j denotes the Pearson correlation coefficient between X(i) and X( j) , X(i) = (X1,i , X2,i , · · · , Xn,i )′ . The strong limit theorem is derived by using Chen-Stein Poisson approximation method.