Abstract

Galton–Watson branching processes bounded from below by a barrier m are considered. These processes become extinct while hitting the states $r = 0,1,2, \ldots ,m$. The extinction probabilities $q_{mr}^{(n)} (t)$, of such a process up to the moment t at the point r, are presented in the form of some finite sums (2) provided this process starts from the state n. For the case $m = 1$ and $r = 0,1$ the extinction probabilities $q_{1r}^{(n)} = \lim _{t \to \infty } q_{1r}^{(n)} (t)$ are written as the sum of series (16). The asymptotic behavior of the probabilities $q_{1r}^{(n)} $ is studied as $n \to \infty $. It is shown that for the subcritical process the probabilities $q_{1r}^{(n)} $ are asymptotically periodic functions in $\log n$ as $n \to \infty $. In the critical case an example is considered in which $q_{1r}^{(n)} $ are given in the form of the series (27); it is shown that in this case $\lim _{n \to \infty } q_{1r}^{(n)} = q_{1r} > 0$, $r = 0,1$.

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