On a limit structure of the Galton–Watson branching processes with regularly varying generating functions

摘要

We investigate limit properties of discrete time branchingprocesses with application of the theory of regularly varying functions inthe sense of Karamata. In the critical situation we suppose that the offspringprobability generating function has an infinite second moment but its tailregularly varies. In the noncritical case, the finite moment of type E[x ln x]is required. The lemma on the asymptotic representation of the generatingfunction of the process and its differential analogue will underlie our conclusions.