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

Let x ₁,…,xₙ be a random sample from a p-dimensional population distribution, where p=pₙ→∞ and log p=o(nβ) for some 0<β≤1, and let Lₙ be the coherence of the sample correlation matrix. In this paper it is proved that √n/log pLₙ→2 in probability if and only if Ee^t₀|x₁₁|α<∞ for some t₀>0, where α satisfies β=α/(4-α). 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.