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On the multifractal analysis of the covering number on the Galton–Watson tree

Part of: Stochastic processes Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  12 July 2019

Najmeddine Attia
Najmeddine Attia*
Affiliation:
Faculté des Sciences de Monastir
*
*Postal address: Department Mathématiques, Faculté des Sciences de Monastir, Avenue de l’Environment 5000, Monastir, Tunisia. Email address: najmeddine.attia@gmail.com
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Abstract

We consider, for t in the boundary of a Galton–Watson tree $(\partial \textsf{T})$, the covering number $(\textsf{N}_n(t))$ by the generation-n cylinder. For a suitable set I and sequence (sn), we almost surely establish the Hausdorff dimension of the set $\{ t \in \partial {\textsf{T}}:{{\textsf{N}}_n}(t) - nb \ {\sim} \ {s_n}\} $ for b ∈ I.

Keywords

Random coveringHausdorff dimensionindexed martingaleGalton–Watson tree

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 265 - 281
Copyright
© Applied Probability Trust 2019 

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