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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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<jpn|∑k=1nXk,iXk,j| and Ln=max 1≤i<jpn|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

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

First available in Project Euclid: 6 March 2006

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Zentralblatt MATH identifier

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

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


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.

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