A functional limit theorem for the logarithm of 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. We prove that, as $n\to\infty$, the sequence of stochastic processes $\{\ln\xi_{[nt]}/(\Delta \sqrt n),\ t\in [0,1]\mid \xi_n>0\}$, where $\Delta$ is some positive constant, converges in distribution to the Brownian excursion $\{W_0^+(t),\ t\in [0,1]\}$ in the space $D[0,1]$ with Skorokhod topology.