Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector
ESAIM: Probability and Statistics, Tome 23 (2019), pp. 797-802.

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$$. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

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Accepté le :
DOI : 10.1051/ps/2019007
Classification : 60J80
Mots-clés : Galton–Watson, branching process, Perron–Frobenius
Cerf, Raphaël 1 ; Dalmau, Joseba 1

1 
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     author = {Cerf, Rapha\"el and Dalmau, Joseba},
     title = {Galton{\textendash}Watson and branching process representations of the normalized {Perron{\textendash}Frobenius} eigenvector},
     journal = {ESAIM: Probability and Statistics},
     pages = {797--802},
     publisher = {EDP-Sciences},
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     year = {2019},
     doi = {10.1051/ps/2019007},
     mrnumber = {4045545},
     zbl = {1506.60091},
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Cerf, Raphaël; Dalmau, Joseba. Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 797-802. doi : 10.1051/ps/2019007. http://archive.numdam.org/articles/10.1051/ps/2019007/

[1] R. Cerf and J. Dalmau, A Markov chain representation of the normalized Perron-Frobenius eigenvector. Electron. Commun. Probab. 22 (2017) 52. | DOI | MR | Zbl

[2] T.E. Harris, The Theory of Branching Processes. Springer-Verlag, Berlin, Prentice-Hall, Inc., Englewood Cliffs, NJ (1963). | DOI | MR | Zbl

[3] E. Seneta, Nonnegative Matrices and Markov Chains, 2nd edn. Springer Series in Statistics. Springer-Verlag, New York (1981). | MR | Zbl

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