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

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dc.creatorLi, Deli
dc.creatorRosalsky, Andrew
dc.date2006-03-14
dc.date.accessioned2026-07-07T07:06:52Z
dc.date.available2026-07-07T07:06:52Z
dc.descriptionLet $\{X_{k,i};i\geq 1,k\geq 1\}$ 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 $W_n=\max_{1\leq i<j\leq p_n}|\sum_{k=1}^nX_{k,i}X_{k,j}|$ and $L_n=\max_{1\leq i<j\leq p_n}|\hatρ^{(n)}_{i,j}|$ where $\hatρ^{(n)}_{i,j}$ denotes the Pearson correlation coefficient between $(X_{1,i},...,X_{n,i})'$ and $(X_{1,j},...,X_{n,j})'$, the limit laws (i) $\lim_{n\to \infty}\frac{W_n}{n^α}=0$ a.s. $(α>1/2)$, (ii) $\lim_{n\to \infty}n^{1-α}L_n=0$ a.s. $(1/2<α\leq 1)$, (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.
dc.descriptionPublished at http://dx.doi.org/10.1214/105051605000000773 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0603334
dc.identifierhttp://arxiv.org/abs/math/0603334
dc.identifierAnnals of Applied Probability 2006, Vol. 16, No. 1, 423-447
dc.identifierdoi:10.1214/105051605000000773
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/110182
dc.subjectProbability
dc.subject60F15, 62H99 (Primary)
dc.titleSome strong limit theorems for the largest entries of sample correlation matrices
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