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

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 $P(\nu =n)$ and $P(\nu >n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $P(\nu =n)$ decays with various speeds such as $\frac{c}{n(\log n{)}^2},$ $\frac{c}{n^{\beta }},\beta >1$ et al. which are very different from the ones of homogeneous multitype Galton-Watson processes.

29 pages; In this version, we prove the results for general mean matrices and some technical conditions are removed

References in corpus (2)

• 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

Cited by in corpus (2)

• On extinction time distribution of a 2-type linear-fractional branching process in a varying environment with asymptotically constant mean matrices
• Asymptotics of entries of products of nonnegative 2-by-2 matrices