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

• 成果类型: Article
• 署名作者: Li, DL; Rosalsky, A
• 署名单位: Lakehead University; State University System of Florida; University of Florida
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051605000000773
• 发表日期: 2006
• 页码: 423-447
• 关键词: sums

摘要

Let (X-k,X-i;i >= 1, k >= 1) be an array of i.i.d. random variables and let {P-n;n >= 1} be a sequence of positive integers such that n/P-n is bounded away from 0 and infinity. For W-n = max1 <= i <= pn vertical bar Sigma(n)(k=1) Xk,iXk,j vertical bar and L-n = max1 <= i,< j <= pn,vertical bar P-i,j((n))vertical bar where p(i,j)((n)) denotes the Pearson correlation coefficient between (XI,i, - - -, Xn,i)' and (Xij, - - -, Xnj)', the limit laws W limn -> oo Wn/n alpha = 0 a.s. (alpha > 1/2), (ii) lim(n ->infinity) n(1-alpha) L-n = 0 a.s. (1/2 < alpha <= 1), (iii) lim(n ->infinity) Wn/root n log n = 2 a.s. and (iv) limn ->infinity (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.