Sojourn times in a finite set of states of Markov branching processes and the probabilities of extinction of a modified Galton–Watson process

In a multi-type branching Galton–Watson process $\mathcal B$, we choose a finite set of states $S$. It is well known that the number of particles $\mu(t)$ at time $t$ in any non-trivial branching process tends with probability one to zero or infinity as $t\to\infty$. Let $\nu_i$ be the number of moments $t$ of discrete time when $\mu(t)$ is equal to the $i$th state of the set $S$. In the first section we prove that the generating function of the multidimensional distribution of $\nu_1,\nu_2,\dots,\nu_r$ is rational. In the second section, for the degenerate Markov branching process $\mathcal B_c$ with particles of one type we find the Laplace transform of the sojourn times $\tau_1,\tau_2,\dots,\tau_r$, or the times of occupation of the states of the set $S=\{1,2,\dots,r\}$. In the third section, we give a method to evaluate the extinction probabilities of a modification $\mathcal B^*$ of the branching process $\mathcal B$.
This research was supported by the Russian Foundation for Basic Research, grants 99–0100012, 00–15–96136, and by INTAS–RFBR, grant 99–01317.