Moment approach for singular values distribution of a large auto-covariance matrix

Abstract

Let (εt)t>0(εt)t>0 be a sequence of independent real random vectors of pp-dimension and let XT=∑s+Tt=s+1εtε∗t−s/TXT=∑t=s+1s+Tεtεt−s∗/T be the lag-ss (ss is a fixed positive integer) auto-covariance matrix of εtεt. Since XTXT is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of XTX∗TXTXT∗. Using the method of moments, we are able to investigate the limiting behaviors of the eigenvalues of XTX∗TXTXT∗ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit FF, which is a result previously developed in (J. Multivariate Anal. 137 (2015) 119–140) using the Stieltjes transform method. Second, we establish the convergence of its largest eigenvalue to the right edge of FF.