Power-fractional distributions and branching processes

TL;DR

The paper develops power-fractional distributions as a targeted generalization of linear-fractional offspring laws for Galton–Watson processes, capturing power-law behavior and closure under iteration. It defines the PF family with parameters $(\theta,\gamma,a,b)$ under cases (A1)–(A3) and MiLe$(\theta)$, and proves a Mittag–Leffler representation of certain sums as a Sibuya mixture of PF variables. For fixed environments, it derives subcritical results including the survival probability limit $\lim_{n\to\infty} a^{n/\theta}\mathbb{P}(Z_n>0) = \left(\frac{a-1}{a+b-1}\right)^{1/\theta}$ and a quasi-stationary PF limit for $Z_n$ conditioned on survival, as well as critical scaling limits toward $\mathsf{CPF}_{+}(\theta,1)$. In random environments, the work provides quenched extinction probabilities, martingale limits $W_\infty$ with explicit Laplace transforms, and a decomposition of supercritical processes into subcritical and explosive components, extending PF concepts to stationary ergodic settings. A continuous-time extension via Sibuya-conjugation of linear birth–death processes yields PF marginals at positive times and a PF Markov branching process framework. Overall, the results deliver explicit, tractable PF models with heavy-tailed behavior applicable to branching systems in varied environments and their continuous-time analogues.

Abstract

In branching process theory, linear-fractional distributions are commonly used to model individual reproduction, especially when the goal is to obtain more explicit formulas than those derived under general model assumptions. In this article, we explore a generalization of these distributions, first introduced by Sagitov and Lindo, which offers similar advantages. We refer to these as power-fractional distributions, primarily because, as we demonstrate, they exhibit power-law behavior. Along with a discussion of their additional properties, we present several results related to the Galton-Watson branching process in both constant and randomly varying environments, illustrating these advantages. The use of power-fractional distributions in continuous time, particularly within the framework of Markov branching processes, is also briefly addressed.