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Limit theorems for multidimensional critical branching processes with immigration. (English. Russian original) Zbl 0541.60089
Sov. Math., Dokl. 28, 220-222 (1983); translation from Dokl. Akad. Nauk SSSR 271, 1066-1069 (1983).
A nondecomposable critical Markov branching process $$\vec z(t)=(z_ 1(t),...,z_ n(t))$$ with n particle types and corresponding generating function $$\vec F(t,s)$$ is submitted to the condition that ($$\vec 1-\vec F(t,s))/(\vec v,\vec 1-\vec F(t,s))\to \vec u$$ uniformly on [0,1) for some positive $$\vec u$$ and $$\vec v$$ with ($$\vec 1,\vec u)=1$$. If $$\vec y(t)$$ is a branching process with immigration whose particles evolve probabilistically like those of $$\vec z(t)$$, then, under a certain regular variation condition, necessary and sufficient criteria for the existence and form of the limit distribution of $$\vec y(t)$$ are presented generalizing earlier results obtained by V. A. Vatutin, Mat. Sb., Nov. Ser. 103(145), 253-264 (1977; Zbl 0365.60079); A. G. Pakes, Adv. Appl. Probab. 11, 31-62 (1979; Zbl 0401.60077) and by the author, Critical branching processes with several particle types and immigration. Teor. Veroyatn. Primen. 27, 348-353 (1982); English translation in Theory Probab. Appl. 27, 369-374 (1982).
Reviewer: F.T.Bruss

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)