Linear fractional Galton-Watson processes in random environment and perpetuities

TL;DR

This paper explores the connections between linear fractional Galton-Watson processes in random environments, random difference equations, and perpetuities, providing an explicit perspective on their behavior and stationary limits, especially in the quenched regime.

Contribution

It offers an explicit view on how random difference equations and perpetuities relate to Galton-Watson processes in random environments, emphasizing the quenched analysis.

Findings

Connection between Galton-Watson processes and random difference equations clarified.

Stationary limits (perpetuities) are characterized in the context of these processes.

The perspective aids understanding of the processes' behavior in the quenched regime.

Abstract

Linear fractional Galton-Watson branching processes in i.i.d.~random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s.~convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.