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

Let {X, X-k.j; i >= 1, k >= 1} be a double array of nondegenerate i.i.d. random variables and let {p(n); n >= 1) be a sequence of positive integers such that nip, is bounded away from 0 and infinity. This paper is devoted to the solution to an open problem posed in Li et al. (2010) [9] on the asymptotic distribution of the largest entry L-n = max(1 <= i<j<pn), vertical bar(rho) over cap ((n))(ij)vertical bar of the (n) sample correlation matrix Gamma(n) = ((rho) over cap ((n))(ij)) where where (rho) over cap ((n))(ij) denotes the Pearson correlation coefficient between (X-1,X-i, ... , X-n,X-i)' and (X-1,X-j, ... , X-n,X-j)'. We show under the assumption EX2 < infinity that the following three statements are equivalent: (1) lim(n ->infinity) n(2) integral(infinity)((n log n)1/4) (Fn-1(x) - Fn-1 (root n log n/x)) dF(x) = 0, (2) (n/log n)(1/2) L-n ->(P) 2, logn)Ln> 2, (3) lim(n ->infinity) P (nL(n)(2) - a(n) <= t) = exp {- 1/root 8 pi (e-t/2)}, -infinity < t < infinity where F(x) = P(vertical bar X vertical bar <= x), x >= 0 and a(n) = 4 log p(n) - log log p(n), n >= 2. To establish this result, we present six interesting new lemmas which may be of independent interest. (C) 2012 Elsevier Inc. All rights reserved.