April 2023 A Kesten–Stigum type theorem for a supercritical multitype branching process in a random environment
Ion Grama, Quansheng Liu, Erwan Pin
Author Affiliations +
Ann. Appl. Probab. 33(2): 1213-1251 (April 2023). DOI: 10.1214/22-AAP1840

Abstract

Consider a multitype branching process in a random environment, whose reproduction law of generation n depends on the random environment at time n, unlike a constant distribution assumed in the Galton–Watson process. The famous Kesten–Stigum theorem for a supercritical multitype Galton–Watson process gives a precise description of the exponential increasing rate of the population size via a criterion for the nondegeneracy of the fundamental martingale. Finding the corresponding result in the random environment case is a longstanding problem. For the single-type case the problem has been solved by Athreya and Karlin for the sufficiency (Ann. Math. Stat. 42 (1971) 1499–1520) and Tanny for the necessity (Stochastic Process. Appl. 28 (1988) 123–139), but for the multitype case it has been open for 50 years. Here we solve this problem in the typical case, by constructing a suitable martingale which reduces to the fundamental one in the constat environment case, and by establishing a criterion for the nondegeneracy of its limit. The convergence in law of the direction of the branching process is also considered. Our results open ways in establishing other limit theorems, such as law of large numbers, central limit theorems, Berry–Essen bound, and large deviation results.

Funding Statement

This work has been supported by the Centre Henri Lebesgue (CHL, ANR-11-LABX-0020- 01), and the National Natural Science Foundation of China (Grant Nos. 11731012, 11971063 and 12271062).

Acknowledgments

We would like to thank the referees and the Associate Editor for their helpful comments and remarks.

Quansheng Liu is the corresponding author.

Citation

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Ion Grama. Quansheng Liu. Erwan Pin. "A Kesten–Stigum type theorem for a supercritical multitype branching process in a random environment." Ann. Appl. Probab. 33 (2) 1213 - 1251, April 2023. https://doi.org/10.1214/22-AAP1840

Information

Received: 1 January 2020; Revised: 1 October 2021; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1515.60275
MathSciNet: MR4564425
Digital Object Identifier: 10.1214/22-AAP1840

Subjects:
Primary: 60J80 , 60K37
Secondary: 60J85

Keywords: Kesten–Stigum theorem , martingale , Multitype branching processes , Products of random matrices , random environment

Rights: Copyright © 2023 Institute of Mathematical Statistics

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Vol.33 • No. 2 • April 2023
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