NECESSARY AND SUFFICIENT CONDITIONS FOR THE ASYMPTOTIC DISTRIBUTIONS OF COHERENCE OF ULTRA-HIGH DIMENSIONAL RANDOM MATRICES

Let x(1),...,x(n) be a random sample from a p-dimensional population distribution, where p = p(n) -> infinity and log p = o(n(beta)) for some 0 < beta <= 1, and let L-n be the coherence of the sample correlation matrix. In this paper it is proved that root n/log pL(n) -> 2 in probability if and only if Ee(t0 vertical bar x11 vertical bar alpha) < infinity for some t(0) > 0, where alpha satisfies beta = alpha/(4 - alpha). Asymptotic distributions of L-n are also proved under the same sufficient condition. Similar results remain valid for m-coherence when the variables of the population are m dependent. The proofs are based on self-normalized moderate deviations, the Stein-Chen method and a newly developed randomized concentration inequality