Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Prof. Quansheng Liu,Univ. Bretagne Sud, France
Inviter:	Professor Xingbiao Hu
Title:	Large deviations for supercritical multi-type branching processes in random environments
Language:	Chinese
Time & Venue:	2023.07.29 10:00-11:00 N613
Abstract:	Consider a $d$-type supercritical branching process $Z_n^i= (Z_n^i(1), \cdots, Z_n^i(d)), n \geq 0,$ in an independent and identically distributed random environment $\xi =(\xi_0, \xi_1, \ldots)$, starting with one initial particle of type $i$, whose offspring distributions of generation $n$ depend on the environment $\xi_n$ at time $n$. I will present a result on the precise large deviations for the total population size $\\|Z_n\\|_1: = \sum_{j=1}^n Z_n(j)$ of generation $n$. The proof is based on the Kesten-Stigum type theorem about the limit of the normalized population size $Z_n^{i}(j)/E_\xi Z_n^i(j)$ (where $E_\xi$ denotes the conditional expectation given the environment $\xi$), the $L^p$ convergence results of this sequence, and precise large deviations for the products of random matrices. The talk is based on a joint work with I. Grama and T. T. Nguyen.