Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices

Let x1,.,xn be a random sample from a p-dimensional population distribution, where p=pn.. and logp=o(n.) for some 0<..1, and let Ln be the coherence of the sample correlation matrix. In this paper it is proved that .n/logpLn.2 in probability if and only if Eet0|x11|.<. for some t0>0, where . satisfies .=./(4..). Asymptotic distributions of Ln 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.