A Note on the Limiting Spectral Distribution of a Symmetrized Auto-Cross Covariance Matrix

TL;DR

This paper presents an alternative approach to deriving the limiting spectral distribution of a symmetrized auto-cross covariance matrix, relaxing the moment conditions needed compared to previous methods.

Contribution

It introduces a new method based on Bai and Silverstein's result, requiring only finite second moments, thus broadening the applicability of the spectral distribution analysis.

Findings

Derived LSD under weaker moment conditions

Simplified proof using Bai and Silverstein's framework

Broader applicability to matrices with finite second moments

Abstract

In Jin et al. (2014), the limiting spectral distribution (LSD) of a symmetrized auto-cross covariance matrix is derived using matrix manipulation, with finite $(2+\delta)$-th moment assumption. Here we give an alternative method using a result in Bai and Silverstein (2010), in which a weaker condition of finite 2nd moment is assumed.