Subcritical multitype branching process in random environment

TL;DR

This paper investigates the long-term survival probabilities and distribution of particles in a multitype branching process within a random environment, focusing on the strongly subcritical case and providing new asymptotic and limit results.

Contribution

It introduces new asymptotic formulas for survival probabilities and a conditional limit theorem for particle distribution in multitype branching processes in random environments.

Findings

Survival probability decays exponentially in the strongly subcritical case.

Conditional distribution of particles converges to a specific limit.

Provides a detailed asymptotic analysis of the process.

Abstract

We study the asymptotic behaviour of the survival probability of a multitype branching process in random environment. The class of processes we consider here corresponds, in the one-dimensional situation, to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.