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

Abstract

Suppose $X$ is an $N\times n$ complex matrix whose entries are centered, independent, and identically distributed random variables with variance $1/n$ and whose fourth moment is of order $\mathcal{O}(n^{-2})$. In the first part of the paper, we consider the non-Hermitian matrix $XA{X}^{\ast }-z$, where $A$ is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and $z\ne 0$ is a complex number. Asymptotic probability bounds for the smallest singular value of this model are obtained in the large dimensional regime where $N$ and $n$ diverge to infinity at the same rate. In the second part of the paper, we consider the special case where $A=J=[1_{i-j=1\mathrm{mod}n}]$ is a circulant matrix. Using the result of the first part, it is shown that the limit eigenvalue distribution of $XJ{X}^{\ast }$ exists in the large dimensional regime, and we determine this limit explicitly. A statistical application of this result devoted towards testing the presence of correlations within a multivariate time series is considered. Assuming that $X$ represents a ${\mathbb{C}}^N$-valued time series which is observed over a time window of length $n$, the matrix $XJ{X}^{\ast }$ represents the one-step sample autocovariance matrix of this time series. Guided by the result on the limit spectral measure of this matrix, a whiteness test against an MA correlation model on the time series is introduced. Numerical simulations show the excellent performance of this test.

43 pages, 6 figures

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• Products of independent elliptic random matrices
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Cited by in corpus (5)

• The asymptotic distribution of the condition number for random circulant matrices
• Singular value distribution of dense random matrices with block Markovian dependence
• Universality of the least singular value for the sum of random matrices
• On eigenvalue distributions of large auto-covariance matrices
• Spectral measure of empirical autocovariance matrices of high dimensional Gaussian stationary processes