Conditioned multi-type Galton-Watson trees

TL;DR

This paper characterizes the distribution of multi-type Galton-Watson trees under various recursive conditioning events, revealing that the conditioned trees retain a multi-type structure with explicitly defined offspring distributions.

Contribution

It provides explicit formulas for the distribution of conditioned multi-type Galton-Watson trees, extending understanding of their structure under complex conditioning.

Findings

Conditioned trees remain multi-type Galton-Watson trees.

Explicit offspring distribution formulas are derived.

The methods apply to a wide range of recursive conditioning events.

Abstract

We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more types and with offspring distributions, depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.