High level subcritical branching processes in a~random environment

V. I. Afanasyev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 10-21

Résumé

A subcritical branching process in a random environment is considered under the assumption that the moment-generating function of a step of the associated random walk $\Theta(t)$, $t\geq0$, is equal to 1 for some value of the argument $\varkappa>0$. Let $T_x$ be the time when the process first attains the half-axis $(x,+\infty)$ and $T$ be the lifetime of this process. It is shown that the random variable $T_x/\ln x$, considered under the condition $T_x+\infty$, converges in distribution to a degenerate random variable equal to $1/\Theta'(\varkappa)$, and the random variable $T/\ln x$, considered under the same condition, converges in distribution to a degenerate random variable equal to $1/\Theta'(\varkappa)-1/\Theta'(0)$.