Limiting Spectral Distribution of Block Matrices with Toeplitz Block Structure

TL;DR

This paper investigates the limiting spectral distributions of symmetric block Toeplitz matrices with different block structures, revealing conditions under which they converge to known laws like the semicircle law.

Contribution

It establishes the existence and form of the LSDs for block Toeplitz matrices with i.i.d. or Toeplitz blocks under various asymptotic regimes, connecting to known spectral laws.

Findings

LSD exists under various asymptotic regimes

In case (i), the LSD converges to the semicircle law

In case (ii), the LSD relates to previous Toeplitz matrix results

Abstract

We study two specific symmetric random block Toeplitz (of dimension $k \times k$) matrices: where the blocks (of size $n \times n$) are (i) matrices with i.i.d. entries, and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by $\sqrt{nk}$) when (a) $k$ is fixed and $n \to\infty$ (b) $n$ is fixed and $k\rightarrow \infty$ (c) $n$ and $k$ go to $\infty$ simultaneously. Further the LSD's obtained in (a) and (b) coincide with those in (c) when $n$ or respectively $k$ tends to infinity. This limit in (c) is the semicircle law in case (i). In Case (ii) the limit is related to the limit of the random symmetric Toepiltz matrix as obtained by Bryc et al.(2006) and Hammond and Miller(2005).