The Annals of Applied Probability

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

Deli Li and Andrew Rosalsky

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Abstract

Let {Xk,i;i≥1,k≥1} be an array of i.i.d. random variables and let {pn;n≥1} be a sequence of positive integers such that n/pn is bounded away from 0 and ∞. For Wn=max 1≤i<j≤pn|∑k=1nXk,iXk,j| and Ln=max 1≤i<j≤pn|ρ̂(n)i,j| where ρ̂(n)i,j denotes the Pearson correlation coefficient between (X1,i,…,Xn,i)' and (X1,j,…,Xn,j)', the limit laws (i) $\lim_{n\rightarrow \infty}\frac{W_{n}}{n^{\alpha}}=0$ a.s. (α>1/2), (ii) lim n→∞n1−αLn=0 a.s. (1/2<α≤1), (iii) $\lim_{n\rightarrow \infty}\frac{W_{n}}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\rightarrow \infty}(\frac{n}{\log n})^{1/2}L_{n}=2$ 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.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 1 (2006), 423-447.

Dates
First available in Project Euclid: 6 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1141654293

Digital Object Identifier
doi:10.1214/105051605000000773

Mathematical Reviews number (MathSciNet)
MR2209348

Zentralblatt MATH identifier
1098.60034

Subjects
Primary: 60F15: Strong theorems 62H99: None of the above, but in this section

Keywords
Largest entries of sample correlation matrices strong law of large numbers law of the logarithm almost sure convergence

Citation

Li, Deli; Rosalsky, Andrew. Some strong limit theorems for the largest entries of sample correlation matrices. Ann. Appl. Probab. 16 (2006), no. 1, 423--447. doi:10.1214/105051605000000773. https://projecteuclid.org/euclid.aoap/1141654293


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