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

AbstractLet {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 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 Ln=max1≤i<j≤pn|ρˆi,j(n)| of the sample correlation matrix Γn=(ρˆi,j(n))1≤i,j≤pn where ρˆi,j(n) denotes the Pearson correlation coefficient between (X1,i,…,Xn,i)′ and (X1,j,…,Xn,j)′. We show under the assumption EX2<∞ that the following three statements are equivalent: (1)limn→∞n2∫(nlogn)1/4∞(Fn−1(x)−Fn−1(nlognx))dF(x)=0,(2)(nlogn)1/2Ln→P2,(3)limn→∞P(nLn2−an≤t)=exp{−18πe−t/2},−∞<t<∞ where F(x)=P(|X|≤x),x≥0 and an=4logpn−loglogpn, n≥2. To establish this result, we present six interesting new lemmas which may be of independent interest