Galton-Watson Probability Contraction

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 $1\le j\le m$, 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 $\Psi ={\Psi }_c:D\to D$ 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 $c\le 1$ and $c>1$, though the techniques for these two ranges are quite different.

This revised version includes two figures, and more detailed proofs of some theorems and lemmas