Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	刘全升教授, 法国Université Bretagne-Sud
Inviter:
Title:	Multitype branching processes in random environments
Language:	Chinese
Time & Venue:	2023.03.08 14:00-15:00 南楼N202
Abstract:	We consider a branching process with a random environment, in which the reproduction law of generation $n$ depends on the random environment at time n, unlike a constant distribution assumed in the classical Galton-Watson process. The famous Kesten-Stigum theorem on a supercritical multi-type Galton-Watson process gives a precise description of the exponential increasing rate of the population size via a criterion for the non-degeneracy of the fundamental branching martingale. Finding the corresponding result in the random environment case is a longstanding problem. For the single-type case the problem has been solved by Athreya and Kaplin (1971) and Tanny (1988), but for the multi-type case the problem remained open for about 50 years. In this lecture I will explain how to solve this problem, by constructing a suitable martingale which reduces to the fundamental branching martingale in the constat environment case, and by establishing a criterion for the non-degeneracy of its limit. I will then explain how to use this result and limit theorems on products of random matrices to obtain corresponding results on branching processes, such as law of large numbers, central limit theorems, Berry-Essen bound, large and moderate deviation results. (Mainly based on a joint work with Ion Grama and Erwan Pin, Ann. Appl. Prob. 2023.)