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

Let {X-k,X-i;k >= 1,i >= 1}be an array of random variables,{X-k;k >= 1}be a strictly stationary phi-mixing sequence, where X-k=(X-k,X-1,X-k,X-2,<middle dot><middle dot><middle dot>,X-k,p). Let{pn;n >= 1}be a sequence of positive integers such that 0<c(1 )<= p(n)/n(tau) <= c(2)<infinity, where tau >0,c(2 )>= c(1)>0. In this paper, we obtain a strong limit the-orem of L-n = max(1 <= i<j <= pn)|rho(i j)|, where rho(i j) denotes the Pearson correlation coefficient between X(i) and X(j),X(i)=(X-1,i,X-2,i,<middle dot><middle dot><middle dot>,X-n,X-i)'. The strong limit theorem is derived by using Chen-Stein Poisson approximation method