On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime

Abstract

Let $(\varepsilon_{t})_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T=\sum_{t=s+1}^{s+T}\varepsilon_t\varepsilon^T_{t-s}/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of $\varepsilon_t$. This paper investigates the limiting behavior of the singular values of $X_T$ under the so-called {\em ultra-dimensional regime} where $p\to\infty$ and $T\to\infty$ in a related way such that $p/T\to 0$. First, we show that the singular value distribution of $X_T$ after a suitable normalization converges to a nonrandom limit $G$ (quarter law) under the forth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of $G$. Both results are derived using the moment method.