## The Annals of Probability

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

#### 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

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