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

@article{Cerf2019GaltonWatsonAB,
  title={Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector},
  author={Rapha{\"e}l Cerf and Joseba Dalmau},
  journal={ESAIM: Probability and Statistics},
  year={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. 
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