One Explicitly Solvable Model For The Galton-Watson Processes In the Random Environment

TL;DR

This paper introduces an explicitly solvable model for Galton-Watson processes in random environments using fractional linear generating functions, providing explicit distribution formulas and classification results.

Contribution

It presents a new solvable model with explicit distribution calculations and classifications for Galton-Watson processes in random environments.

Findings

Explicit formulas for the distribution of particle numbers over time.

Classification of processes based on the fractional linear generating functions.

Limit theorems for quenched and annealed types.

Abstract

In this paper, we study the Galton-Watson process in the random environment for the particular case when the number of the offsprings in each generation has the fractional linear generation function with random parameters. In this case, the distribution of $N_t$, the number of particles at the moment time $t=0,1,2,\cdots$ can be calculated explicitly. We present the classification of such processes and limit theorems of two types: quenched type which is for the fixed realization of the random environment and annealed type which includes the averaging over the environment.