Multitype branching processes evolving in i.i.d. random environment: probability of survival for the critical case

Using the annealed approach we investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in i.i.d. random environment. 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 type $i$ is equivalent to $c_{i}n^{-1/2}$ for large $n,$ where $c_{i}$ is a positive constant. Earlier such asymptotic representation was known only for the case of the fractional linear offspring generating functions.