Multitype weakly subcritical branching processes in random environment

A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment $X$ of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions $\mathbf{E}X<0$ and $\mathbf{E}Xe^{X}>0$.