A double exponential law for maximal branching processes

We consider maximal branching processes defined by the recurrence relation
$$ Z_{n+1}=\bigvee_{m=1}^{Z_n}\xi_{m,n}, $$
where $\vee$ stands for the operation of taking maximum, $\xi_{m,n}$, $m\ge 1$, $n\ge 0$, are independent with distribution function $F$ on $\mathbf Z_+$.
We prove limit theorems for stationary distributions of the processes $\{Z^{(N)}_n\}$ with the distribution functions $F^{(N)}(x)=F^N(x)$ as $N\to\infty$ in the case where $F$ belongs to the domain of attraction of the double exponential law.
This research was supported by the Russian Foundation for Basic Research, grant 00–01–00131.