On extinction time distribution of a 2-type linear-fractional branching process in a varying environment with asymptotically constant mean matrices

2021/06/02 by Huaming Wang, Wang, Hua-Ming
Mathematics · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2106.01203

Abstract

In this paper we study a 2-type linear-fractional branching process in varying environment with asymptotically constant mean matrices. Let ν be the extinction time and for k≥1 let M_k be the mean matrix of offspring distribution of individuals of the (k-1)-th generation. Under certain conditions, we show that P(ν=n) and P(ν>n) are asymptotically equivalent to some functions of products of spectral radii of the mean matrices. This paper complements a former result [arXiv: 2007.07840] which requires in addition a condition ∀ k≥1,\rmdet(M_k)0. Such a condition excludes a large class of mean matrices. As byproducts, we also get some results on asymptotics of products of nonhomogeneous matrices which have their own interests.

Citations

• On a maximum of nearest-neighbor random walk with asymptotically zero drift on lattice of positive half line

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