Extinction probabilities for branching processes bounded from below

Galton–Watson branching processes bounded from below by a barrier $m$ are considered. These processes become extinct while hitting the states $r=0,1,2,\dots,m$. 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 $\lim_{n\to\infty}q_{1r}^{(n)}=q_{1r}>0$, $r=0,1$.