---
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.'
author:
- Wang Qinwen
- Yao Jianfeng
date: 'Received: date / Accepted: date'
title: 'Moment approach for singular values distribution of a large auto-covariance matrix'
---

Introduction
============

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 $\varepsilon_t$. The motivation of the above set up is due to the study of dynamic factor model, see [@LY012]. Set $$\begin{aligned}
x_t=\Lambda f_t+\varepsilon_t+\mu~,\end{aligned}$$ where $x_t$ is a $p$-dimensional sequence observed at time $t$, $\{f_t\}$ 