A subcritical decomposable branching process in a mixed environment

A two-type decomposable branching process is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. The reproduction law of the first type particles is specified by a random environment. The reproduction law of the second type particles is one and the same for all generations. A limit theorem is proved describing the conditional distribution of the number of particles in the process at time $nt,t\in (0,1]$, given the survival of the process up to moment $n\rightarrow \infty .$