Limit theorems for functionals of a branching process in random environment starting with a large number of particles

Assume given a sequence $Z^{(k)}=Ż_{i}^{(k)}, i=0,1,…\}$, $k=1,2,…$, of critical branching processes in random environment which are only different from one another by the size $k$ of the founding generation. Assume that the step of the associated random walk belongs to the domain of attraction of a stable law. In the case $k=k(n)$, where $n$ is a positive integer parameter and $k(n)$ grows with $n$ in a certain special way, limit theorems as $n\to\infty$ are established for a process with continuous time constructed from $Z^{(k(n))}$ and for the logarithm of this process. In addition, limit theorems are proved for the moment of degeneration of the process $Z^{(k(n))}$, the maximum of this process, and the total number of particles.

critical branching process in a random medium, limit theorems, functional limit theorems