Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang

Abstract

Abstract : In the theory of large random matrices, how to dominate the norm of a random matrix is a very important problem. This paper considers a different type of random matrices, namely -W to the k power, i.e. a power of a square random matrix with iid entries. The first result in this paper is the limit as n approaches infinity of the absolute value of (W/sq rt. n) to the k power is or = (1+k)(sigma to the k power) where n is the size of W and here sigma-sq. is the variance of the entries of W. We assume only the existence of the 4-th moment of the entries of W. From this result it is easy to show that the spectral radius of W sq rt n is not greater then -sigma with probability 1. This result is known only for iid N(O,-sigma-sq) case. In proving the above result, a new kind of graphs has to be discussed carefully, and the truncation method used in Yin-Bai-Krishnaiah is also important here.