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.
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