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

Let {X, X _k,i;i≥1, k≥} be a double array of nondegenerate i.i.d. random variables and let {p _n;n≥} be a sequence of positive integers such that n/p _n is bounded away from 0 and ∞. This paper is devoted to the solution to an open problem posed in Li etal. (2010) [9] on the asymptotic distribution of the largest entry Ln=max1≤i<j≤pn{pipe}ρ̂i,j(n){pipe} of the sample correlation matrix Γn=(ρ̂i,j(n))1≤i,j≤pn where ρ̂ _i,j ⁽ⁿ⁾ _1≤ijpn where ρ̂ _ij ⁽ⁿ⁾ denotes the Pearson correlation coefficient between (X _1,i,..., X _n,i) ^' and (X _1,j,..., X _n,j) ^'. We show under the assumption EX2<∞ that the following three statements are equivalent: (1)lim _n→∞n ²∫ _(nlogn)1/4 ^∞(F ^n-1(x)-F ^n-1(√nlogn/x))dF(x)=0,(2)(n/logn) ^1/2L _n→P{double-struck}2, (3)lim _n→∞P{double-struck}(nL _n ²-a _n≤t)=exp{-1/√8πe ^-t/2},-∞<t<∞ where F(x)=P{double-struck}({pipe}X{pipe}≤x),x≥0 and a _n=4logp _n-loglogp _n, n≥2. To establish this result, we present six interesting new lemmas which may be of independent interest.