Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Abstract

Suppose is an complex matrix whose entries are centered, independent, and identically distributed random variables with variance and whose fourth moment is of order . Suppose is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and is a complex number. First we consider the matrix , and obtain asymptotic probability bounds for its smallest singular value when and diverge to infinity and . Then we consider the special case where is a circulant matrix. Using the above result, we show that the limit spectral distribution of exists when and describe the limit explicitly. Assuming that represents a -valued time series which is observed over a time window of length , the matrix represents the one-step sample autocovariance matrix of this time series. A whiteness test against an MA correlation model for this time series is introduced based on the above limit result. Numerical simulations show the excellent performance of this test.