A class of subcritical branching processes with immigration and infinite number of types of particles

We consider a subcritical branching process with immigration, infinite number of types $T_1,T_2,\ldots$ of particles, and discrete time. The state of the process at the moment of time $t$ is the set of vectors
$$ \vec{\xi}(r,t)=(\xi_1(t),\xi_2(t),\dots,\xi_r(t)), \qquad r\ge1, $$
where $\xi_i(t)$ is the number of particles of type $T_i$ at the moment of time $t$, $i=1,2,\ldots$ It is assumed that at each moment of time only particles of type $T_1$ immigrate and each particle of type $T_i$ turns into a set of particles of types $T_i$ and $T_{i+1}$. It is proved that the probability distributions of the vectors $\vec{\xi}(r,t)$ converge as $t\to\infty$ to discrete limit distributions.