2023/07/06 by Tan, Jiangrui, Zhang, Mei
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2307.02981
This paper is concerned with the characterizations of fixed points of the generating function of branching processes with countably infinitely many types. We assume each particle of type i can only give offspring of type j≥ i, whose number only depends on j-i. We prove that, for these processes, there are at least countably infinitely many fixed points of the offspring generating function, while the extinction probability set of the process has only 2 elements. This phenomenon contrasts sharply with those of finite-type branching processes. Our result takes one step forward on the related conjecture on the fixed points of infinite-dimensional generating functions in literature. In addition, the asymptotic behavior of the components of fixed point is given.