Open Access
May 2007 Exact Hausdorff measure on the boundary of a Galton–Watson tree
Toshiro Watanabe
Ann. Probab. 35(3): 1007-1038 (May 2007). DOI: 10.1214/009117906000000629


A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton–Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions ϕ according to whether ϕ-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes’s conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved.


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Toshiro Watanabe. "Exact Hausdorff measure on the boundary of a Galton–Watson tree." Ann. Probab. 35 (3) 1007 - 1038, May 2007.


Published: May 2007
First available in Project Euclid: 10 May 2007

zbMATH: 1127.60083
MathSciNet: MR2319714
Digital Object Identifier: 10.1214/009117906000000629

Primary: 28A78 , 60J80
Secondary: 28A80 , 60G18

Keywords: b-decomposable distribution , Boundary , branching measure , dominated variation , exact Hausdorff measure , Galton–Watson tree , shift self-similar additive random sequence

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • May 2007
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