## Abstract

Let {X, X _{k,i};i≥1, k≥} be a double array of nondegenerate i.i.d. random variables and let {p _{n};n≥} be a sequence of positive integers such that n/p _{n} is bounded away from 0 and ∞. This paper is devoted to the solution to an open problem posed in Li etal. (2010) [9] on the asymptotic distribution of the largest entry Ln=max1≤i<j≤pn{pipe}ρ̂i,j(n){pipe} of the sample correlation matrix Γn=(ρ̂i,j(n))1≤i,j≤pn where ρ̂ _{i,j} ^{(n)} _{1≤ijpn} where ρ̂ _{ij} ^{(n)} denotes the Pearson correlation coefficient between (X _{1,i},..., X _{n,i}) ^{'} and (X _{1,j},..., X _{n,j}) ^{'}. We show under the assumption EX2<∞ that the following three statements are equivalent: (1)lim _{n→∞}n ^{2}∫ _{(nlogn)1/4} ^{∞}(F ^{n-1}(x)-F ^{n-1}(√nlogn/x))dF(x)=0,(2)(n/logn) ^{1/2}L _{n}→P{double-struck}2, (3)lim _{n→∞}P{double-struck}(nL _{n} ^{2}-a _{n}≤t)=exp{-1/√8πe ^{-t/2}},-∞<t<∞ where F(x)=P{double-struck}({pipe}X{pipe}≤x),x≥0 and a _{n}=4logp _{n}-loglogp _{n}, n≥2. To establish this result, we present six interesting new lemmas which may be of independent interest.

Original language | English (US) |
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Pages (from-to) | 256-270 |

Number of pages | 15 |

Journal | Journal of Multivariate Analysis |

Volume | 111 |

DOIs | |

State | Published - Oct 2012 |

### Bibliographical note

Funding Information:The authors are grateful to the Referees for their constructive, perceptive, and substantial comments and suggestions which enabled them to greatly improve the paper. The authors are also grateful to Dr. Wei-Dong Liu for his interest in their work and for offering some helpful comments. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-1005345 and NSA Grant H98230-10-1-0161 .

Copyright:

Copyright 2012 Elsevier B.V., All rights reserved.

## Keywords

- Asymptotic distribution
- Largest entries of sample correlation matrices
- Law of the logarithm
- Pearson correlation coefficient
- Second moment problem