On singular value distribution of large-dimensional autocovariance matrices

Abstract

Let ${\left({\epsilon }_j\right)}_{j\ge 0}$ be a sequence of independent $p$-dimensional random vectors and $\tau \ge 1$ a given integer. From a sample ${\epsilon }_1,\dots ,{\epsilon }_{T+\tau }$ of the sequence, the so-called lag-$\tau$ auto-covariance matrix is $C_{\tau }=T^{-1}{\sum }_{j=1}^T{\epsilon }_{\tau +j}{\epsilon }_j^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_{\tau }$ assuming that $p$ and $T$ grow to infinity proportionally and the sequence has uniformly bounded $(4+\delta )$th order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix $C_{\tau }$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem.