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On the multifractal spectrum of the branching measure on a Galton–Watson tree

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Peter Mörters  and
Narn-Rueih Shieh
Peter Mörters*
Affiliation:
University of Bath
Narn-Rueih Shieh*
Affiliation:
National Taiwan University and NCTS, Taipei
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, National Taiwan University, Taipei, Taiwan. Email address: [email protected]
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Abstract

Suppose that μ is the branching measure on the boundary of a supercritical Galton–Watson tree with offspring distribution N such that E[N log N] < ∞ and P{N = 1} > 0. We determine the multifractal spectrum of μ using a method different from that proposed by Shieh and Taylor, which is flawed.

Keywords

Galton–Watson treebranching measurebranching setpercolationHausdorff dimensiondimension spectrummultifractal spectrum

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G17: Sample path properties
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1223 - 1229
Copyright
Copyright © Applied Probability Trust 2004 

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References

Bass, R. F. (1995). Probabilistic Techniques in Analysis. Springer, New York.Google Scholar
Dubuc, M. S. (1971). Le densité de la loi-limite d'un processus en cascade expansif. Z. Wahrscheinlichkeitsth. 19, 281–290.Google Scholar
Hawkes, J. (1981). Trees generated by a simple branching process. J. London Math. Soc. 24, 373–384.Google Scholar
Khoshnevisan, D., Peres, Y., and Xiao, Y. (2000). Lim, sup random fractals. Electron. J. Prob. 5, No. 4.CrossRefGoogle Scholar
Lyons, R. (1990). Random walks and percolation on trees. Ann. Prob. 18, 931–958.Google Scholar
Mörters, P., and Shieh, N.-R. (2002). Thin and thick points for branching measure on a Galton–Watson tree. Statist. Prob. Lett. 58, 13–22.Google Scholar
Peres, Y. (1999). Probability on trees: an introductory climb. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997; Lecture Notes Math. 1717), ed. Bertoin, J., Springer, Berlin, pp. 193–280.CrossRefGoogle Scholar
Shieh, N.-R., and Taylor, S. J. (2002). Multifractal spectra of branching measure on a Galton–Watson tree. J. Appl. Prob. 39, 100–111.Google Scholar