On limiting spectral distribution and joint convergence of some patterned random matrices

TL;DR

This paper provides shorter, alternative proofs for the limiting spectral distribution and joint convergence of reverse circulant and symmetric circulant matrices with independent entries, using the moment method.

Contribution

It offers concise, alternative proofs for known results on spectral properties of patterned random matrices, enhancing understanding and methodological simplicity.

Findings

Confirmed limiting spectral distribution for reverse circulant matrices

Established joint convergence for symmetric circulant matrices

Proposed a simplified proof technique based on moments

Abstract

This article deals with the limiting spectral distribution and joint convergence of reverse circulant and symmetric circulant matrices with independent entries. These results are already proved in articles Bose and Sen (2008) \cite{bose_sen_LSD_EJP}, and Bose, Hazra and Saha (2011) \cite{bose_saha_patter_JC_annals}, respectively. But this article provides alternate proofs of these results which are shorter than the existing proofs. Our method is mainly based on moment method.