Reduced branching processes in random environment: the critical case

V. A. Vatutin

Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 1, pp. 21-38

Résumé

Let $Z_n$ be the number of particles at time $n=0,1,2,\dots$ in a branching process in random environment, $Z_0=1$, and let $Z_{m,n}$ be the number of such particles in the process at time $m\in[0,n]$, each of which has a nonempty offspring at time $n$. It is shown that if the offspring generating functions $f_k(s)$ of the particles of the $k$th generation are independent and identically distributed for all $k=0,1,2,\dots$ with $E\log f'_k(1)=0$ and $\sigma^2=E(\log f'_k(1))^2\in(0,\infty)$, then, under certain additional restrictions, the sequence of conditional processes $$ \biggl\{\frac1{\sigma\sqrt{n}}\,\log Z_{[nt],n},\,t\in[0,1]\bigm|Z_n>0\biggr\} $$ converges, as $n\to\infty$, in distribution in Skorokhod topology to the process $\{\inf_{t\le u\le 1}W^+(u),\,t\in[0,1]\}$, where $\{W_+(t),\,t\in [0,1]\}$ is the Brownian meander.

Détail

Keywords: critical branching process in random environment, reduced process, functional limit theorem, random walk.