A new limit theorem for a critical branching process in a random environment

Let $\{\xi_n\}$ be a critical branching process in random environment with linear-fractional generating functions, $m_n$ be the conditional expectation of $\xi_n$ with respect to random environment. We prove a theorem on convergence of the sequence of random processes $\{\xi_{[nt]}/m_{[nt]},\ t\in(0,1] \mid \xi_n>0\}$ as $n\to\infty$ in distribution in the corresponding functional space.