Badalbaev, I. S. Limit theorems for multivariate branching processes with immigration of increasing intensity. (Russian) Zbl 0627.60079 Dokl. Akad. Nauk UzSSR 1983, No. 2, 3-5 (1983). 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 Cited in 2 Documents MSC: 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) Keywords:asymptotic behavior; Galton-Watson branching process; immigration; Perron-Frobenius eigenvalue; slowly varying function PDFBibTeX XMLCite \textit{I. S. Badalbaev}, Dokl. Akad. Nauk UzSSR 1983, No. 2, 3--5 (1983; Zbl 0627.60079)