A branching process with migration in a random environment

We study a Galton–Watson branching process $\{Z_n\}_{n=0}^\infty$ with migration in a random environment which is specified by a stationary Markov chain $\{\eta_n\}_{n=0}^\infty$ with finite state space. Let $f_{\eta_n}(z)$ be the offspring generating function of each particle of the $n$th generation, $M=\lim_{n\to\infty}\mathsf E\log f_{\eta_n}'(1)$.
It is proved that the stationary distribution of the properly normalized number of particles in the process $\{Z_n\}_{n=0}^\infty$ converges to the uniform distribution on the interval $[0,1]$ as $M\to 1$.
The work was supported by the Russian Foundation for Basic Research, grant 96–01–00338 and INTAS–RFBR 95–0099.