Galton-Watson Probability Contraction

TL;DR

This paper studies the probabilities of first order statements in Galton-Watson trees with Poisson offspring, using Ehrenfeucht games and contraction mappings to analyze probability distributions across different regimes.

Contribution

It introduces a novel recursive framework and contraction mapping approach to analyze probability distributions of equivalence classes in Galton-Watson trees for all offspring means.

Findings

Contraction mapping ensures convergence to true probability distribution.

Different techniques are used for regimes c ≤ 1 and c > 1.

Fixed point characterization of class probabilities in Galton-Watson trees.

Abstract

We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with $Poisson(c)$ offspring distribution. Fixing a positive integer $k$, we exploit the $k$-move Ehrenfeucht game on rooted trees for this purpose. Let $\Sigma$, indexed by $1 \leq j \leq m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $\Sigma$. Let $x_{j}(c)$ denote the true probability of the class $j \in \Sigma$ under $Poisson(c)$ regime, and $\vec{x}(c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $\Gamma$, and a map $\Psi = \Psi_{c}: D \rightarrow D$ such that $\vec{x}(c)$ is a fixed point of $\Psi_{c}$, and starting with any distribution $\vec{x} \in D$, we converge to this fixed point via $\Psi$ because it is a contraction. We show this both for $c \leq 1$ and $c > 1$, though the techniques for these two ranges are quite different.