Some strong limit theorems for the largest entries of sample correlation matrices

Details Some strong limit theorems for the largest entries of sample correlation matrices Some strong limit theorems for the largest entries of sample correlation matrices

2006-03-14

arxiv.org/abs/math/0603334v1

Annals of Applied Probability 2006, Vol. 16, No. 1, 423-447

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051605000000773 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051605000000773

Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i1/2)$, (ii) $\lim_{n\to \infty}n^{1-\alpha}L_n=0$ a.s. $(1/2<\alpha \leq 1)$, (iii) $\lim_{n\to \infty}\frac{W_n}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\to \infty}(\frac{n}{\log n})^{1/2}L_n=2$ a.s. are shown to hold under optimal sets of conditions. These results follow from some general theorems proved for arrays of i.i.d. two-dimensional random vectors. The converses of the limit laws (i) and (iii) are also established. The current work was inspired by Jiang's study of the asymptotic behavior of the largest entries of sample correlation matrices.

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