Asymptotically Pseudo-Independent Matrices

TL;DR

This paper demonstrates that a specific family of pseudo-random matrices exhibits asymptotic independence, with different sequences generated from distinct primitive polynomials becoming independent as size increases.

Contribution

It establishes the asymptotic independence property for a family of pseudo-random matrices previously introduced by Soloveychik, Xiang, and Tarokh.

Findings

Different matrix sequences become asymptotically independent

Asymptotic independence holds for matrices from different primitive polynomials

Supports the pseudo-random matrices' utility in applications requiring independence

Abstract

We show that the family of pseudo-random matrices recently discovered by Soloveychik, Xiang, and Tarokh in their work `Symmetric Pseudo-Random Matrices' exhibits asymptotic independence. More specifically, any two sequences of matrices of matching sizes from that construction generated using sequences of different non-reciprocal primitive polynomials are asymptotically independent.