Seminars## A tutorial course on Marchenko-Pastur Law in random matrix theory: (1) Wishart matrix

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Chu-Chi Lee

2013-12-13

12:30:00 - 15:00:00

103 , Mathematics Research Center Building (ori. New Math. Bldg.)

This 3-lecture (6 hours) tutorial course will go over the proofs of Marchenko-Pastur Law by Stieltjes transform. Random-matrix theory has become a major tool in many fields, including number theory and combinatorics, wireless communications (Tulino and Verdu, 2004), and in multivariate statistical analysis and principal components analysis (Johnstone, 2001).In this tutorial, we will concentrate on the Marchenko–Pastur’s Quarter-Circle Law when both r and n go to infinity. Here the random matrix is of size rxr and n is the number of observations. An example of such a random matrix is the sample covariance matrix which is often encountered in Statistics such as PCA. When the matrix aspect ratio r/n goes to a non-zero constant, the empirical distribution of the eigenvalues follows Marchenko-Pastur law.

References: Z. Bai and J. W. Silverstein, \"Spectral Analysis of Large Dimensional Random Matrices,\" Springer. Marcenko, V.A, and Pastur, L.A. (1967). Distributions of eigenvalues of some sets of random matrices, Math. USSR-Sb, 1, 507–536.