Binary Galton-Watson trees with mutations

TL;DR

This paper analyzes a multitype Galton-Watson process with mutations, providing explicit calculations of type distributions over time and long-term proportions without relying on approximations.

Contribution

It introduces a method for exact computation of type distributions and mutation timings in multitype Galton-Watson trees, extending previous models.

Findings

Explicit formulas for mean and distribution of types over time

Estimation of long-term type proportions

Distribution of first occurrence times for specific types

Abstract

We consider a multitype Galton-Watson process that allows for the mutation and reversion of individual types in discrete and continuous time. In this setting, we explicitly compute the time evolution of quantities such as the mean and distributions of different types. This allows us in particular to estimate the proportions of different types in the long run, as well as the distribution of the first time of occurrence of a given type as the tree size or time increases. Our approach relies on the recursive computation of the joint distribution of types conditionally to the value of the total progeny. In comparison with the literature on related multitype models, we do not rely on approximations.