Necessary and sufficient conditions for a nonnegative matrix to be strongly R-positive

TL;DR

This paper characterizes when a nonnegative matrix can be associated with a Markov chain having exponential return time moments, linking spectral radius behavior to matrix entry modifications.

Contribution

It provides a concise, largely self-contained proof that strong R-positivity is equivalent to the spectral radius decreasing when finitely many entries are lowered.

Findings

Strong R-positivity is characterized by exponential moments of return times.

Lowering finitely many entries of the matrix decreases the spectral radius.

The paper offers a simplified proof of this equivalence.

Abstract

Using the Perron-Frobenius eigenfunction and eigenvalue, each finite irreducible nonnegative matrix $A$ can be transformed into a probability kernel $P$. This was generalized by David Vere-Jones who gave necessary and sufficient conditions for a countably infinite irreducible nonnegative matrix $A$ to be transformable into a recurrent probability kernel $P$, and showed uniqueness of $P$. Such $A$ are called R-recurrent. Let us say that $A$ is strongly R-positive if the return times of the Markov chain with kernel $P$ have exponential moments of some positive order. Then it is known that strong R-positivity is equivalent to the property that lowering the value of finitely many entries of $A$ lowers the spectral radius. This paper gives a short and largely self-contained proof of this fact.