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

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

2004-04-02

arxiv.org/abs/math/0404053v1

Ann. Probab., 23, 1102 - 1124 (1995)

Mathematics

Probability

30 pages

Scientific paper

Evans defines 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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