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

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 $k\ge1$ 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 $\forall k\ge1,\rm{det}(M_k)<-\varepsilon$ for some $\varepsilon>0.$ 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.
Comment: 22 pages