Geodesic
A class of subcritical branching processes with immigration and infinite number of types of particles
Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 6-10.

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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. 
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B. A. Sevast'yanov. A class of subcritical branching processes with immigration and infinite number of types of particles. Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 6-10. http://geodesic.mathdoc.fr/item/DM_2007_19_1_a1/

[1] Sevastyanov B. A., Vetvyaschiesya protsessy, Nauka, Moskva, 1971

[2] Kesten H., Stigum B. P., “A limit theorem for multidimensional Galton–Watson processes”, Ann. Math. Statist., 37:5 (1966), 1211–1223 | DOI | MR | Zbl