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 $R_{\tau }^{ϵ}$ 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 $R_{\tau }^{ϵ}$, 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 $R_{\tau }^{ϵ}$ 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 $R_{\tau }^{ϵ}$ 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.