Galton-Watson processes and their role as building blocks for branching processes

TL;DR

This paper explores the Galton-Watson process as a fundamental tool in branching processes, highlighting its applications, extensions, and significance in various complex models and real-world problems.

Contribution

It provides an expository and argumentative overview of Galton-Watson processes, including new extensions like bisexual and resource-dependent models, emphasizing their versatility and relevance.

Findings

Galton-Watson processes serve as versatile building blocks for complex branching models.

Extensions such as bisexual and resource-dependent processes enhance modeling capabilities.

The paper illustrates applications in medicine and societal processes.

Abstract

This article is an essay, both expository and argumentative, on the Galton-Watson process as a tool in the domain of Branching Processes. It is at the same time the author's ways to honour two distinguished scientists in this domain, both from the Russian Academy of Science, and to congratulate them for their special birthdays coming up very soon. The thread of the article is the role, which the Galton-Watson process had played in the author's own research. We start with an article on a controlled Galton-Watson process. Then we pass to random absorbing processes, and also recall and discuss a problem in medicine. Further questions will bring us via the Borel-Cantelli Lemma to $\varphi$-branching processes and extensions. To gain more generality, we then look at bisexual Galton-Watson processes. Finally we briefly discuss relatively complicated resource dependent branching processes to show that, here again, using Galton-Watson reproduction schemes (whenever reasonable) can be a convincing approach to new processes which are then sufficiently tractable to obtain results of interest. Keywords: Controlled branching process; $\varphi$-branching process, Bisexual reproduction, Borel-Cantelli Lemma; Resource dependence; Society forms, Stopping times, Theorem of envelopment, BRS-inequality.