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An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree
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  • Published: January 2000

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

  • Steven P. Lalley1 &
  • Thomas Sellke2 

Probability Theory and Related Fields volume 116, pages 41–56 (2000)Cite this article

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  • 3 Citations

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Abstract.

Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2−n where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ωμ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ−1 (Ωμ) in the metric d is

almost surely on the event of nonextinction, where h(μ) is the entropy of the measure μ and q(i, j) is the mean number of type-j offspring of a type-i individual. This extends a theorem of HAWKES [5], which shows that the Hausdorff dimension of the entire boundary at infinity is log2 α, where α is the Malthusian parameter.

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Authors and Affiliations

  1. University of Chicago, Department of Statistics, Chicago, IL 60637, USA. e-mail: lalley@stat.purdue.edu, , , , , , US

    Steven P. Lalley

  2. Purdue University, Department of Statistics, West Lafayette, IN 47907, USA. e-mail: tsellke@stat.purdue.edu, , , , , , US

    Thomas Sellke

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  1. Steven P. Lalley
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  2. Thomas Sellke
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Received: 30 June 1998 / Revised: 4 February 1999

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Lalley, S., Sellke, T. An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree. Probab Theory Relat Fields 116, 41–56 (2000). https://doi.org/10.1007/PL00008722

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  • Issue Date: January 2000

  • DOI: https://doi.org/10.1007/PL00008722

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  • Mathematics Subject Classification (1991): 60J80
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