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This article is cited in 13 scientific papers (total in 13 papers)
Reduced branching processes in random environment: the critical case
V. A. Vatutin Steklov Mathematical Institute, Russian Academy of Sciences
Abstract:
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.
Keywords:
critical branching process in random environment, reduced process, functional limit theorem, random walk.
Received: 27.08.2001
Citation:
V. A. Vatutin, “Reduced branching processes in random environment: the critical case”, Teor. Veroyatnost. i Primenen., 47:1 (2002), 21–38; Theory Probab. Appl., 47:1 (2003), 99–113
Linking options:
https://www.mathnet.ru/eng/tvp2959https://doi.org/10.4213/tvp2959 https://www.mathnet.ru/eng/tvp/v47/i1/p21
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