## The Annals of Applied Probability

### The asymptotic distributions of the largest entries of sample correlation matrices

Tiefeng Jiang

#### Abstract

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

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

Dates
First available in Project Euclid: 23 April 2004

https://projecteuclid.org/euclid.aoap/1082737115

Digital Object Identifier
doi:10.1214/105051604000000143

Mathematical Reviews number (MathSciNet)
MR2052906

Zentralblatt MATH identifier
1047.60014

#### Citation

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. https://projecteuclid.org/euclid.aoap/1082737115

#### References

• Amosova, N. N. (1972). On limit theorems for the probabilities of moderate deviations. Vestnik Leningrad. Univ. 13 5--14.
• Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, $2$nd ed. Wiley, New York.
• Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximation: The Chen--Stein method. Ann. Probab. 17 9--25.
• Bai, Z. D. (1993). Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21 1275--1294.
• Bai, Z. D. (1999). Methodologies in spectral analysis of large dimensional random matrices, a review. Statist. Sinica 9 611--677.
• Barbour, A. and Eagleson, G., (1984). Poisson convergence for dissociated statistics. J. R. Stat. Soc. Ser. B Stat. Methodol. 46 397--402.
• Chow, Y. S. and Teicher, H. (1988). Probability Theory, Independence, Interchangeability, Martingales, 2nd ed. Springer, New York.
• Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
• Hoffman-Jørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math. 52 159--186.
• Jiang, T. (2002). Maxima of partial sums indexed by geometrical structures. Ann. Probab. 30 1854--1892.
• Jiang, T. (2002). A comparison of scores of two protein structures with foldings. Ann. Probab. 30 1893--1912.
• Jiang, T. (2002). Maxima of entries of Haar distributed matrices. Preprint.
• Jiang, T. (2002). The limiting distributions of eigenvalues of sample correlation matrices. Preprint.
• Johnstone, I. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295--327.
• Ledoux, M. (1992). On moderate deviations of sums of i.i.d. vector random variables. Ann. Inst. H. Poincar$\acute\mboxe$ Probab. Statist. 28 267--280.
• Ledoux, M. and Talagrand, M. (1992). Isoperimetry and Processes in the Theory of Probability in a Banach Space. Springer, New York.
• Li, D., Rao, M., Jiang, T. and Wang, X. (1995). Complete convergence and almost sure convergence of weighted sums of random variables. J. Theoret. Probab. 8 754--789.
• Mheta, M. L. (1991). Random Matrices, 2nd ed. Academic Press, Boston.
• Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745--789.
• Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
• Rubin, H. and Sethuraman, J. (1965). Probabilities of moderate deviations. Sankhyā Ser. A, 325--346.