Galton-Watson and branching process representations of the normalized Perron-Frobenius eigenvector

TL;DR

This paper derives formulas expressing the normalized Perron-Frobenius eigenvector of a primitive matrix using multitype Galton-Watson and branching processes, generalizing classical Markov chain formulas.

Contribution

It introduces new formulas linking Perron-Frobenius eigenvectors to multitype branching processes, extending classical invariant measure formulas.

Findings

Formulas for eigenvector as a functional of Galton-Watson process

Formulas involving multitype branching process with matrix exponential

Generalization of classical Markov chain invariant measure formula

Abstract

Let $A$ be a primitive matrix and let $\lambda$ be its Perron-Frobenius eigenvalue. We give formulas expressing the associated normalized Perron-Frobenius eigenvector as a simple functional of a multitype Galton-Watson process whose mean matrix is $A$, as well as of a multitype branching process with mean matrix $e^{(A-I)t}$. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.