Multitype subcritical branching processes in a random environment

We investigate a multitype Galton–Watson process in a random environment generated by a sequence of independent identically distributed random variables. Assuming that the mean of the increment $X$ of the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices is negative and the random variable $Xe^X$ has zero mean, we find the asymptotics of the survival probability at time $n$ as $n\to\infty$.