Characterization of fixed points of infinite-dimensional generating functions

TL;DR

This paper investigates fixed points of generating functions in infinite-type branching processes, revealing a complex structure with infinitely many fixed points but only two extinction probabilities, advancing understanding in infinite-dimensional stochastic processes.

Contribution

It characterizes the fixed points of infinite-dimensional generating functions for a specific class of branching processes, showing the existence of infinitely many fixed points and analyzing their asymptotic behavior.

Findings

Infinitely many fixed points of the generating function exist.

Extinction probability set has only two elements.

Asymptotic behavior of fixed point components is described.

Abstract

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 $j\geq i$, 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.