Survival probability for a class of multitype subcritical branching processes in random environment

We study the asymptotic behaviour of the survival probability of a multi-type branching processes in random environment. The class of processes we consider corresponds, in the one-dimensional situation, to the intermediately subcritical case. We show under rather general assumptions on the form of the offspring generating functions of particles that the probability of survival up to generation $n$ of the process initiated at moment zero by a single particle of any type is of order $\lambda ^{n}n^{-1/2}$ for large $n,$ where $\lambda \in (0,1)$ is a constant specified by the Lyapunov exponent of the mean matrices of the process.