On singular values of large dimensional lag-τ sample auto-correlation matrices

We study the limiting behavior of singular values of a lag-& tau; sample auto-correlation matrix R & epsilon;& tau; of large dimensional vector white noise process, the error term & epsilon; in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) that characterizes the global spectrum of R & epsilon;& tau;, 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 R & epsilon;& tau; is the same as that of the lag-& tau; sample auto -covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of R & epsilon;& tau; 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-& tau; sample auto-correlation matrices in a factor model. Our theoretical results are fully supported by numerical experiments as well. & COPY; 2023 Elsevier Inc. All rights reserved.