Some limit theorems for the critical Galton–Watson branching processes

Kh.  Kudratov; Ya.  Khusanbaev

doi:10.37863/umzh.v75i4.6781

Kh. Kudratov National University of Uzbekistan, Tashkent
Ya. Khusanbaev Institute of Mathematics, Tashkent, Uzbekistan

DOI: https://doi.org/10.37863/umzh.v75i4.6781

Keywords: Critical Galton-Watson process, generating function, slowly varying function.

Abstract

UDC 519.21

We consider critical Galton–Watson processes starting from a random number of particles and determine the effect of the mean value of the initial state on the asymptotic state of the process. For processes starting from a large number of particles and satisfying the condition $(S),$ we prove the limit theorem similar to the result of W. Feller. We also prove the theorem under the condition $W(n)>0$ for critical processes satisfying the conditions $(S)$ and $(M).$

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