Convergence of high dimensional Toeplitz and related matrices with correlated inputs

TL;DR

This paper studies the joint convergence of high-dimensional Toeplitz and related matrices with correlated inputs, revealing universal limit behaviors independent of entry distributions, and extends to Hankel matrices.

Contribution

It establishes universal joint convergence results for complex and real Toeplitz, Hankel, and related matrices with correlated entries, generalizing previous symmetric cases.

Findings

Joint convergence depends only on correlation structure

Results include asymmetric Hankel matrices as special cases

Limits are universal, independent of entry distributions

Abstract

We investigate the joint convergence of independent random Toeplitz matrices with complex input entries that have a pair-correlation structure, along with deterministic Toeplitz matrices and the backward identity permutation matrix. Further, we study the joint convergence of independent generalized Toeplitz matrices along with other related matrices. The limits depend only on the correlation structure but are universal otherwise, in that they do not depend on the underlying distributions of the entries. In particular, these results provide the joint convergence of asymmetric Hankel matrices. Earlier results in the literature on the joint convergence of random symmetric Toeplitz and symmetric Hankel matrices with real entries follow as special cases.