Exact Hausdorff measure on the boundary of a Galton-Watson tree

• 成果类型: Article
• 署名作者: Watanabe, Toshiro
• 署名单位: University of Aizu
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000629
• 发表日期: 2007
• 页码: 1007-1038
• 关键词: semi-selfdecomposable distributions; additive random sequences; self-similar processes; exact packing measure; branching measure; random fractals; independent increments; continuity properties; large deviations; LIMIT-THEOREMS

摘要

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 phi according to whether pi-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.