Large deviation results for Critical Multitype Galton-Watson trees

TL;DR

This paper establishes a joint large deviation principle for empirical measures of critical multitype Galton-Watson trees conditioned on size, extending previous results to more general offspring laws and providing explicit rate functions.

Contribution

It introduces a joint large deviation principle for empirical measures of critical multitype Galton-Watson trees conditioned on size, including non-Markov offspring laws.

Findings

Derived a large deviation principle for empirical measures in critical multitype Galton-Watson trees.

Extended existing large deviation results to cover non-Markov offspring distributions.

Provided explicit rate functions in terms of relative entropy for specific offspring laws.

Abstract

In this article, we prove a joint large deviation principle in $n$ for the \emph{empirical pair measure} and \emph{ empirical offspring measure} of critical multitype Galton-Watson trees conditioned to have exactly $n$ vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by \cite[Theorem~2.1]{DMS03}. For the case where the offspring law of the tree is a geometric distribution with parameter \sfrac{1}{2}, we get an exact rate function. All our rate functions are expressed in terms of relative entropies.