Fixed point theorem for an infinite Toeplitz matrix

TL;DR

This paper establishes conditions for the existence of nontrivial bounded positive solutions to the fixed point equation involving an infinite Toeplitz matrix with nonnegative entries, relevant to stochastic processes and recurrence relations.

Contribution

It provides new criteria for solutions of fixed point equations for infinite Toeplitz matrices, linking matrix properties to asymptotic behavior in stochastic and applied contexts.

Findings

Derived conditions for nontrivial solutions

Linked matrix properties to recurrence relation behavior

Applicable to stochastic process analysis

Abstract

For an infinite Toeplitz matrix $T$ with nonnegative real entries we find the conditions, under which the equation $\boldsymbol{x}=T\boldsymbol{x}$, where $\boldsymbol{x}$ is an infinite vector-column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behavior of convolution type recurrence relations, and can be applied to different problems arising in the theory of stochastic processes and applied problems from other areas.