The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment

The asymptotic behavior of the survival probability for multi-type branching processes in a random environment is studied. In the case where all particles are of one type, the class of processes under consideration corresponds to intermediately subcritical processes. Under fairly general assumptions on the form of the generating functions of the laws of reproduction of particles, it is proved that the survival probability at a remote instant $n$ of time for a process that started at the zero instant of time from one particle of any type is of the order of $\lambda^{n}n^{-1/2}$, where $\lambda \in (0,1)$ is a constant defined in terms of the Lyapunov exponent for products of the mean-value matrices of the laws of reproduction of particles.