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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 21 open journal systems 


Another look at the moment method for large dimensional random matrices

Arup Bose, Indian Statistical Institute
Arnab Sen, University of California, Berkeley


Abstract
The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the  well known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample variance covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) establish the LSD for the random Toeplitz and Hankel matrices using the moment method.  They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalent classes and relating the limits of the counts to certain volume calculations.

We build on their work and present a unified approach. This helps provide  relatively short and easy proofs for the LSD of common matrices while at the same time providing insight into the nature of different LSD and their interrelations. By extending these methods we are also able to deal with matrices with appropriate dependent entries.


Full text: PDF

Pages: 588-628

Published on: April 12, 2008
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Electronic Journal of Probability. ISSN: 1083-6489