On singular values of large dimensional lag-[formula omitted] sample auto-correlation matrices

Abstract

We study the limiting behavior of singular values of a lag-τ sample auto-correlation matrix Rτϵ of large dimensional vector white noise process, the error term ϵ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) that characterizes the global spectrum of Rτϵ, and derive the limit of its largest singular value. All the asymptotic results are derived under the high-dimensional asymptotic regime where the data dimension and sample size go to infinity proportionally. Under mild assumptions, we show that the LSD of Rτϵ is the same as that of the lag-τ sample auto-covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of Rτϵ converges almost surely to the right end point of the support of its LSD. Based on these results, we further propose two estimators of total number of factors with lag-τ sample auto-correlation matrices in a factor model. Our theoretical results are fully supported by numerical experiments as well.