Limit theorems for a moderately subcritical branching process in a random environment

Let $\{\xi_n\}$ be a moderately subcritical branching process in a random environment with linear-fractional generating functions, $m_n$ be the conditional expectation of $\xi_n$ with respect to the random environment. We prove theorems 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, and of the initial and final segments of the random sequence $\xi_0/m_0,\xi_1/m_1,\ldots,\xi_n/m_n$ considered under the condition that $\{\xi_n>0\}$.