Abstract
In this paper, we consider the limit properties of the largest entries of sample covariance matrices and the sample correlation matrices. In order to make the statistics based on the largest entries of the sample covariance matrices and the sample correlation matrices more applicable in high-dimensional tests, the identically distributed assumption of population is removed. Under some moment’s assumption of the underlying distribution, we obtain that the almost surely limit and asymptotical distribution of the extreme statistics as both the dimension p and sample size n tend to infinity.
Highlights
Where xi (1/n) nk 1 xik. en, Rn: (ρij) is the p × p sample correlation matrix generated by Xn
Let Tn and Ln denote the largest entries of off-diagonal elements of Sn and Rn, respectively, that is, Tn 1≤mi
We mainly study the asymptotic properties of Tn, Ln
Summary
Under above assumptions (i)–(iii), we have that log nTn 2 a.s.. Jiang [6] has proved that eorems 1 and 2 hold if xik are i.i.d. random variables under some moment conditions. Under above assumptions (i)–(iii), we have that log nLn 2 a.s.. Cai and Jiang [15] investigate the limit properties of Ln for p ≫ n for i.i.d. random variables under more stronger moment assumption. We first introduce some lemmas that are used for the proofs of main variables results.
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