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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

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