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Limiting Spectral Distributions of Large Dimensional Random Matrices

Models where the number of parameters increases with the sample size, are becoming increasingly important in statistics. This necessitates a close look at the statistical properties of eigenvalues of random matrices whose dimension increases indefinitely. There are several properties of the eigenvalues that one would be interested in and the literature in this area is already huge. In this article we focus on one important aspect: the existence and identification of the limiting spectral distribution (LSD) of the empirical distribution of the eigenvalues. We describe some of the general tools used in establishing the LSD and how they have been applied successfully to establish results on the LSD for certain types of matrices. Some of the matrices for which the LSD has been established and the nature of the limit laws known are described in detail. We also discuss a few open problems and partial solutions for some of these. We introduce a few new ideas which seem to hold some promise in this area. We also establish an invariance result for random Toeplitz matrix.