Limiting spectral distribution of Toeplitz and Hankel matrices with dependent entries

TL;DR

This paper investigates the spectral distribution limits of symmetric Toeplitz and Hankel matrices with dependent entries formed from sums of i.i.d. variables, providing explicit moments and an alternative proof for classical cases.

Contribution

It introduces explicit moment formulas for LSDs of Toeplitz and Hankel matrices with dependent entries and extends the moment method to other patterned matrices.

Findings

Explicit moment sequences for LSDs derived

Alternative proof for classical i.i.d. entry case provided

Method applicable to reverse and symmetric circulant matrices

Abstract

This article deals with the limiting spectral distributions (LSD) of symmetric Toeplitz and Hankel matrices with dependent entries. For any fixed positive integer $m$, we consider these $n \times n$ matrices with entries $\{Y^{(m)}_j / \sqrt n \ ;{j\geq 0}\}$, where $Y^{(m)}_j= \sum_{r=-m}^{m} X_{j+r}$ and $\{X_k\}$ are i.i.d. with mean zero and variance one. We provide an explicit expression for the moment sequences of the LSDs. As a special case, this article provides an alternate proof for the LSDs of these matrices when the entries are i.i.d. with mean zero and variance one. The method is based on the moment method. The idea of proof can also be applied to other patterned random matrices, namely reverse circulant and symmetric circulant matrices.