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Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix
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  • Published: 06 May 2009

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

  • Deli Li1,
  • Wei-Dong Liu2 &
  • Andrew Rosalsky3 

Probability Theory and Related Fields volume 148, pages 5–35 (2010)Cite this article

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  • 23 Citations

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Abstract

Let {X k,i; i ≥ 1, k ≥ 1} be a double array of nondegenerate 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 ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry \({L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}\) of the sample correlation matrix \({{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}\) where \({\hat{\rho}^{(n)}_{i,j}}\) denotes the Pearson correlation coefficient between (X 1,i , ..., X n,i)′ and (X 1,j ,...,X n,j)′. Write \({F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}\) , \({W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}\) , and \({W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}\) . Under the assumption that \({\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}\) for some δ > 0, we show that the following six statements are equivalent:

$$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$
$$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$
$$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$
$$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$
$$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$
$$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$

where \({\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}\) , and a n  = 4 log p n − log log p n . The equivalences between (i), (ii), (iii), and (v) assume that only \({\mathbb{E}X_{1,1}^{2} < \infty}\) . Weak laws of large numbers for W n and L n , n ≥  1, are also established and these are of the form \({W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)\) and \({n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)\), respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L n obtained by Jiang. Some open problems are also posed.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, P7B 5E1, Canada

    Deli Li

  2. Department of Mathematics, Zhejiang University, 310027, Hangzhou, China

    Wei-Dong Liu

  3. Department of Statistics, University of Florida, Gainesville, FL, 32611, USA

    Andrew Rosalsky

Authors
  1. Deli Li
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  2. Wei-Dong Liu
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  3. Andrew Rosalsky
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Correspondence to Andrew Rosalsky.

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Li, D., Liu, WD. & Rosalsky, A. Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probab. Theory Relat. Fields 148, 5–35 (2010). https://doi.org/10.1007/s00440-009-0220-z

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  • Received: 06 March 2008

  • Revised: 24 February 2009

  • Published: 06 May 2009

  • Issue Date: September 2010

  • DOI: https://doi.org/10.1007/s00440-009-0220-z

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Keywords

  • Asymptotic distribution
  • Largest entries of sample correlation matrices
  • Weak law of large numbers
  • Weak law of the logarithm

Mathematics Subject Classification (2000)

  • Primary: 60F05
  • 60F10
  • Secondary: 62H99
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