The Annals of Probability

Galton-Watson Trees with the Same Mean Have the Same Polar Sets

Robin Pemantle and Yuval Peres

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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.

Article information

Source
Ann. Probab., Volume 23, Number 3 (1995), 1102-1124.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988175

Digital Object Identifier
doi:10.1214/aop/1176988175

Mathematical Reviews number (MathSciNet)
MR1349163

Zentralblatt MATH identifier
0833.60085

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G60: Random fields

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

Citation

Pemantle, Robin; Peres, Yuval. Galton-Watson Trees with the Same Mean Have the Same Polar Sets. Ann. Probab. 23 (1995), no. 3, 1102--1124. doi:10.1214/aop/1176988175. https://projecteuclid.org/euclid.aop/1176988175


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