Asymptotic distribution of the largest entry of sample correlation matrices from 1-dependent normal population

Abstract

In this paper, we investigate the limiting distribution of the largest entry of sample correlation matrices in the ultra-high-dimension case. Let M_{n,p} = (X_{k,i}) be an n × p random matrix whose n rows are n observations coming from the p-dimensional 1-dependent normal population. Suppose both the population dimension p and the sample size n are large with log p = o(n^{t}) for any t ∈ (0, 1/3]. For L_{n} = max_{1≤i<j≤p}ρ_{ij}, where ρ_{ij} is the Pearson correlation coefficient between i-th column and j-th column of M_{n,p}, the limiting distribution and the law of large numbers are obtained by the Chen–Stein Poisson approximation method and the moderation deviation principle. And an example is given to test the covariance structure.