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样本相关阵最大元的几个强极限定理

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

Deli Li, Andrew Rosalsky

备注：
Published 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)
发表信息：
Annals of Applied Probability 2006, Vol. 16, No. 1, 423-447
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摘要：

令$\{X_{k,i};i\geq 1,k\geq 1\}$是i.i.d.随机变量，并让$\{p_n;n\geq 1\}$是正整数序列，使得$n/p_n$有界远离0和$\infty$。对于$W_n=\max_{1\leq i<j\leq p_n}|\sum_{k=1}^nX_{k,i}X_{k,j}|$和$L_n=\max_{1\leq i<j\leq p_n}|\hatρ^{(n)}_{i,j}|$，其中$\hatρ^{(n)}_{i,j}$表示$(X_{1,i},...,X_{n,i})'$和$(X_{1,j},...,X_{n,j})'$之间的Pearson相关系数，极限定律（i）$\lim_{n\to \infty}\frac{W_n}{n^α}=0$ a.s.（ii）$\lim_{n\to \infty}n^{1-α}L_n=0$ a.s.（iii）$\lim_{n\to \infty}\frac{W_n}{\sqrt{n\log n}}=2$ a.s.和（iv）$\lim_{n\to \infty}(\frac{n}{\log n})^{1/2}L_n=2$ a.s.在最佳条件下成立。这些结果遵循一些一般定理证明的阵列的i.i.d.二维随机向量建立了极限定律（i）和（iii）的逆式。当前的工作受到了Jiang对样本相关阵的最大项的渐近行为的研究的启发。

英文摘要：

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 $W_n=\max_{1\leq i<j\leq p_n}|\sum_{k=1}^nX_{k,i}X_{k,j}|$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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