On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix

Let {X,Xk,i; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {pn; n ≥ 1} be a sequence of positive integers such that n/pn is bounded away from 0 and ∞. This work is devoted to the solution to an open problem posed in Li, Liu, and Rosalsky (2010) on the asymptotic distribution of the largest entry Ln = max1≤i<j≤pn ∣∣ρ̂(n) i,j ∣∣ of the sample correlation matrix Γn = ( ρ̂ (n) i,j ) 1≤i,j≤pn where ρ̂ (n) i,j denotes the Pearson correlation coefficient between (X1,i, · · · , Xn,i)′ and (X1,j, · · · , Xn,j)′. We show under the assumption EX < ∞ that the following three statements are equivalent: