Moment approach for singular values distribution of a large auto-covariance matrix

TL;DR

This paper investigates the limiting distribution of singular values of large auto-covariance matrices using moment methods, establishing convergence of the spectral distribution and the largest eigenvalue.

Contribution

It introduces a moment approach to analyze the eigenvalues of large auto-covariance matrices, providing new theoretical results on their spectral behavior.

Findings

Empirical spectral distribution converges to a nonrandom limit.

Largest eigenvalue converges to the right edge of the limit.

Results are derived using moment methods.

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 $\varepsilon_t$. Since $X_T$ is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of $X_TX^T_T$. Therefore, the purpose of this paper is to investigate the limiting behaviors of the eigenvalues of $X_TX^T_T$ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit $F$. Second, we establish the convergence of its largest eigenvalue to the right edge of $F$. Both results are derived using moment methods.