Probabilities of extinction of subcritical multitype branching processes that are bounded from below

An initial multitype Galton–Watson branching process
$$ \mu(t)=(\mu_1(t),\dots,\mu_m(t)), \qquad t=0,1,2,\dots, $$
generates a bounded from below branching process if at the moment when $\mu(t)$ enters some finite set $S$ the process stops. We study a bounded from below subcritical Galton–Watson branching process $\xi(t)=(\xi_1(t),\dots,\xi_m(t))$ with $m$ types of particles $T_1,\dots,T_m$ whose absorbing states form a set $S=\{0,e(j_1),\dots,e(j_{m_1})\}$, where $e(j)=(\delta_{j1},\dots,\delta_{jm})$, $1\le m_1\le m$, and 0 is the zero vector. It is shown that
$$ q_j^n=\lim_{t\to\infty}\mathsf P\{\xi(t)=e(j)\mid \xi(0)=n\}, $$
where $n=(n_1,\dots,n_m)$, converges, as $\bar n=n_1+\ldots+n_m\to\infty$ and $n_i/\bar n\to a_i$, to some periodic with period 1 function of $\log_{1/R}\bar n$, where $R<1$ is the Perron root of the matrix of the mathematical expectations of the initial branching process.
This work was supported by the Russian Foundation for Basic Research, grant 96–01–00338, and INTAS–RFBR 95-0099.