The Annals of Applied Probability

The asymptotic distributions of the largest entries of sample correlation matrices

Tiefeng Jiang

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Let Xn=(xij) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let Rn=(ρij) be the p×p sample correlation matrix of Xn; that is, the entry ρij is the usual Pearson”s correlation coefficient between the ith column of Xn and jth column of Xn. For contemporary data both n and p are large. When the population is a multivariate normal we study the test that H0: the p variates of the population are uncorrelated. A test statistic is chosen as Ln=max ijij|. The asymptotic distribution of Ln is derived by using the Chen–Stein Poisson approximation method. Similar results for the non-Gaussian case are also derived.

Article information

Ann. Appl. Probab., Volume 14, Number 2 (2004), 865-880.

First available in Project Euclid: 23 April 2004

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 62H10: Distribution of statistics

Sample correlation matrices maxima Chen–Stein method moderate deviations


Jiang, Tiefeng. The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 (2004), no. 2, 865--880. doi:10.1214/105051604000000143.

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