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

Let $\{\xi_n\}$ be a critical branching process in a random environment, and let $m_n$ be the mathematical expectation of $\xi_n$ under the condition that the random environment is fixed. We prove a theorem on convergence of the sequence of branching processes $\{\xi_{[nt]}/m_{[nt]},\ t\in(0,1] \mid \xi_n>0\}$ as $n\to\infty$ in distribution in the corresponding functional space. This theorem extends the earlier result of the author proved under the assumption that the generating function of the number of offspring is linear-fractional.