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Limit theorems for multivariate branching processes with immigration of increasing intensity. (Russian) Zbl 0627.60079

The r-type Galton-Watson branching process admits immigration of particles from the environment. Three theorems are provided (without proof), for the Perron-Frobenius eigenvalue of the expected progeny matrix, less than, equal to, or greater than one.
Under various moment assumptions it is shown, essentially, that if the i- th component of the expected immigration behaves as \(\lambda^ n_ i\) \(n^{\gamma_ i}L(n)\), where n is time, \(\lambda_ i\), \(\gamma_ i>0\) are constants and L(n) is a slowly varying function, then the vector \([(Z_ 1-EZ_ 1)/DZ_ 1,...,(Z_ r-EZ_ r)/DZ_ r]\), tends to an r- dimensional Gaussian law. The covariance matrix has nondiagonal elements with moduli less than one.
Reviewer: M.Kimmel

MSC:

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