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Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices. (English) Zbl 1354.60020
Summary: 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 \(0<\beta\leq 1\), and let \(L_{n}\) be the coherence of the sample correlation matrix. In this paper it is proved that \(\sqrt{n/\log p}L_{n}\to 2\) 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.

MSC:
60F05 Central limit and other weak theorems
60B20 Random matrices (probabilistic aspects)
60F10 Large deviations
62E20 Asymptotic distribution theory in statistics
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