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

¹ Département de Mathématiques et Applications, Ecole Normale Supérieure, CNRS, PSL Research University, 75005 Paris, France.
² Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France.
³ Centre de Mathématiques Appliquées, Ecole Polytechnique, CNRS, Université Paris-Saclay, 91405 Orsay, France.

^* Corresponding author: joseba.dalmau@polytechnique.edu.

Received: 11 September 2018
Accepted: 1 April 2019

Abstract

Let A be a primitive matrix and let λ 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.