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Two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices

Part of: Special matrices Markov processes

Published online by Cambridge University Press:  07 February 2022

Hua-Ming Wang  and
Huizi Yao
Hua-Ming Wang*
Affiliation:
Anhui Normal University
Huizi Yao*
Affiliation:
Anhui Normal University & Suzhou University
*
*Postal address: School of Mathematics & Statistics, Anhui Normal University, Wuhu, 241003, China
**Postal address: Faculty of Mathematics & Statistics, Suzhou University, Suzhou, 234000, China
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Abstract

Consider two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices. Let $\nu$ be the extinction time. Under certain conditions, we show that both $\mathbb{P}(\nu=n)$ and $\mathbb{P}(\nu>n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $\mathbb{P}(\nu=n)$ decays with various speeds such as ${c}/({n^{1/2}\log n)^2}$ , ${c}/{n^\beta}$ , $\beta >1$ , which are very different from those of homogeneous multitype Galton–Watson processes.

Keywords

Branching processextinction timeproduct of non-negative matricesspectral radiustail of continued fraction

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 15B48: Positive matrices and their generalizations; cones of matrices
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 224 - 255
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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