General linear-fractional branching processes with discrete time

TL;DR

This paper analyzes a general linear-fractional branching process with a broad type space, deriving explicit formulas and limit theorems that extend previous results to more complex, multi-type scenarios.

Contribution

It introduces a new framework for linear-fractional branching processes with a general type space and derives explicit distribution formulas and limit theorems.

Findings

Explicit distribution formulas for arbitrary observation times.

Limit theorems for subcritical, critical, and supercritical cases.

Extension of previous results to processes with a general type space.

Abstract

We study a linear-fractional Bienaym\'e-Galton-Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads to the linear-fractional distribution formula for an arbitrary observation time, which allows us to establish transparent limit theorems for the subcritical, critical and supercritical cases. Our results extend recent findings for the linear-fractional branching processes with countably many types.