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July, 1995 Galton-Watson Trees with the Same Mean Have the Same Polar Sets
Robin Pemantle, Yuval Peres
Ann. Probab. 23(3): 1102-1124 (July, 1995). DOI: 10.1214/aop/1176988175

Abstract

Evans defined a notion of what it means for a set $B$ to be polar for a process indexed by a tree. The main result herein is that a tree picked from a Galton-Watson measure whose offspring distribution has mean $m$ and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies that deterministic and nondeterministic trees behave identically in a variety of probability models. Mapping subsets of Euclidean space to trees and polar sets to capacity criteria, it follows that certain random Cantor sets are capacity-equivalent to each other and to deterministic Cantor sets. An extension to branching processes in varying environment is also obtained.

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Robin Pemantle. Yuval Peres. "Galton-Watson Trees with the Same Mean Have the Same Polar Sets." Ann. Probab. 23 (3) 1102 - 1124, July, 1995. https://doi.org/10.1214/aop/1176988175

Information

Published: July, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0833.60085
MathSciNet: MR1349163
Digital Object Identifier: 10.1214/aop/1176988175

Subjects:
Primary: 60J80
Secondary: 60D05 , 60G60 , 60J45

Keywords: branching , capacity , Galton-Watson , percolation , polar sets , random Cantor sets , tree

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.23 • No. 3 • July, 1995
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