On singular value distribution of large dimensional auto-covariance matrices

TL;DR

This paper analyzes the asymptotic behavior of the singular value distribution of large-dimensional auto-covariance matrices derived from dependent data sequences, extending random matrix theory to more complex, non-symmetric cases.

Contribution

It establishes the limiting singular value distribution of large auto-covariance matrices with dependent entries, introducing new techniques to handle non-symmetry and dependence.

Findings

Derived the limit of singular value distribution for large auto-covariance matrices.

Extended random matrix theory to dependent, non-symmetric matrices.

Provided new mathematical techniques for analyzing complex auto-covariance structures.

Abstract

Let $(\varepsilon_j)_{j\geq 0}$ be a sequence of independent $p-$dimensional random vectors and $\tau\geq1$ a given integer. From a sample $\varepsilon_1,\cdots,\varepsilon_{T+\tau-1},\varepsilon_{T+\tau}$ of the sequence, the so-called lag $-\tau$ auto-covariance matrix is $C_{\tau}=T^{-1}\sum_{j=1}^T\varepsilon_{\tau+j}\varepsilon_{j}^t$. 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 satisfies a Lindeberg condition on fourth 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.