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Using the annealed approach we investigate the asymptotic behavior of thesurvival probability of a critical multitype branching process evolving ini.i.d. random environment. We show under rather general assumptions on the formof the offspring generating functions of particles that the probability ofsurvival up to generation $n$ of the process initiated at moment zero by asingle 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 representationwas known only for the case of the fractional linear offspring generatingfunctions.

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