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2017 Galton-Watson probability contraction
Moumanti Podder, Joel Spencer
Electron. Commun. Probab. 22: 1-16 (2017). DOI: 10.1214/17-ECP47

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.

Citation

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Moumanti Podder. Joel Spencer. "Galton-Watson probability contraction." Electron. Commun. Probab. 22 1 - 16, 2017. https://doi.org/10.1214/17-ECP47

Information

Received: 19 February 2016; Accepted: 23 January 2017; Published: 2017
First available in Project Euclid: 16 March 2017

zbMATH: 1360.60157
MathSciNet: MR3627009
Digital Object Identifier: 10.1214/17-ECP47

Subjects:
Primary: 05C20‎
Secondary: 60F20

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