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Ladder variables, internal structure of Galton–Watson trees and finite branching random walks

Part of: Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Jean-François Marckert  and
Abdelkader Mokkadem
Jean-François Marckert*
Affiliation:
Université de Versailles
Abdelkader Mokkadem*
Affiliation:
Université de Versailles
*
∗Postal address: Université de Versailles, 45 Avenue des Etats Unis, 78035 Versailles Cedex, France
∗Postal address: Université de Versailles, 45 Avenue des Etats Unis, 78035 Versailles Cedex, France
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Abstract

In this paper, we consider Galton–Watson trees conditioned by size. We show that the number of k-ancestors (ancestors that have k children) of a node u is (almost) proportional to its depth. The k, j-ancestors are also studied. The methods rely on the study of ladder variables on an associated random walk. We also give an application to finite branching random walks.

Keywords

Ladder variablesGalton–Watson treesbranching random walks

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 671 - 689
Copyright
Copyright © Applied Probability Trust 2003 

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