Paper detail

Galton-Watson Probability Contraction

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 $Σ$, 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 $Σ$. Let $x_{j}(c)$ denote the true probability of the class $j \in Σ$ 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 $Γ$, and a map $Ψ= Ψ_{c}: D \rightarrow D$ such that $\vec{x}(c)$ is a fixed point of $Ψ_{c}$, and starting with any distribution $\vec{x} \in D$, we converge to this fixed point via $Ψ$ 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.