Reduced multitype critical branching processes in random environment

We consider a multitype critical branching process $\mathbf{Z}_{n},n=0,1,...$, in an i.i.d. random environment. Let $Z_{m,n}$ be the number of particles in this process at time $m$ having descendants at time $n$. A limit theorem is proved for the logarithm of $Z_{nt,n}$ at moments $nt,\,0\leq t\leq 1,$ conditioned on the survival of the process $\mathbf{Z}_{n}$ up to moment $n$ when $n\rightarrow \infty $.