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

Let $\mathbf {x}_1,\ldots,\mathbf {x}_n$ be a random sample from a $p$-dimensional population distribution, where $p=p_n\to\infty$ and $\log p=o(n^{\beta})$ for some $0<\beta\leq1$, and let $L_n$ be the coherence of the sample correlation matrix. In this paper it is proved that $\sqrt{n/\log p}L_n\to2$ in probability if and only if $Ee^{t_0|x_{11}|^{\alpha}}<\infty$ 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.