Volume 23, 2019
|Page(s)||797 - 802|
|Published online||24 December 2019|
Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector
Département de Mathématiques et Applications, Ecole Normale Supérieure, CNRS, PSL Research University,
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: email@example.com.
Accepted: 1 April 2019
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.
Mathematics Subject Classification: 60J80
Key words: Galton–Watson / branching process / Perron–Frobenius
© The authors. Published by EDP Sciences, SMAI 2019
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