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

Abstract

In this paper we study a 2-type linear-fractional branching process in varying environment with asymptotically constant mean matrices. Let $\nu$ be the extinction time and for $k\ge 1$ let $M_k$ be the mean matrix of offspring distribution of individuals of the $(k-1)$-th generation. Under certain conditions, we show that $P(\nu =n)$ and $P(\nu >n)$ 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 $\forall k\ge 1,\det ({\mathrm{M}}_{\mathrm{k}})<-\epsilon$ for some $\epsilon >0.$ 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.

arXiv comment22 pages

References in corpus (3)

• Time-Inhomogeneous Branching Processes Conditioned on Non-Extinction
• On a maximum of nearest-neighbor random walk with asymptotically zero drift on lattice of positive half line
• Two-type linear fractional branching processes in varying environments with asymptotically constant mean matrices

Cited by in corpus (1)

• Regeneration of branching processes with immigration in varying environments