2018/12/18 by Arup Bose, Bose, Arup, Walid Hachem +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1812.07237
openalex publication_date 2018/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose X is an N \× n complex matrix whose entries are centered,\nindependent, and identically distributed random variables with variance 1/n\nand whose fourth moment is of order mathcal O(n-2). In the first part\nof the paper, we consider the non-Hermitian matrix X A X^* - z, where A is\na deterministic matrix whose smallest and largest singular values are bounded\nbelow and above respectively, and z\≠ 0 is a complex number. Asymptotic\nprobability bounds for the smallest singular value of this model are obtained\nin the large dimensional regime where N and n diverge to infinity at the\nsame rate.\n In the second part of the paper, we consider the special case where A = J =\n[1i-j = 1 mod n ] is a circulant matrix. Using the result of the first\npart, it is shown that the limit eigenvalue distribution of X J X^* exists in\nthe large dimensional regime, and we determine this limit explicitly. A\nstatistical application of this result devoted towards testing the presence of\ncorrelations within a multivariate time series is considered. Assuming that X\nrepresents a mathbb CN-valued time series which is observed over a time\nwindow of length n, the matrix X J X^* represents the one-step sample\nautocovariance matrix of this time series. Guided by the result on the limit\nspectral measure of this matrix, a whiteness test against an MA correlation\nmodel on the time series is introduced. Numerical simulations show the\nexcellent performance of this test.\n