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

Arup Bose and Walid Hachem

Journal of Multivariate Analysis, 2020, vol. 178, issue C

Abstract: Suppose X is an N×n complex matrix whose entries are centered, independent, and identically distributed random variables with variance 1∕n and whose fourth moment is of order O(n−2). Suppose A is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and z≠0 is a complex number. First we consider the matrix XAX∗−z, and obtain asymptotic probability bounds for its smallest singular value when N and n diverge to infinity and N∕n→γ,0<γ<∞. Then we consider the special case where A=J=[1i−j=1modn] is a circulant matrix. Using the above result, we show that the limit spectral distribution of XJX∗ exists when N∕n→γ,0<γ<∞ and describe the limit explicitly. Assuming that X represents a ℂN-valued time series which is observed over a time window of length n, the matrix XJX∗ 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.

Keywords: Large non-Hermitian matrix theory; Limit spectral distribution; Smallest singular value; Whiteness test in multivariate time series (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.jmva.2020.104623

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