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Multifractal spectra of branching measure on a Galton-Watson tree

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Narn-Rueih Shieh  and
S. James Taylor
Narn-Rueih Shieh*
Affiliation:
National Taiwan University
S. James Taylor*
Affiliation:
University of Sussex
*
∗ Postal address: Department of Mathematics, National Taiwan University, Taipei, Taiwan. Email address: [email protected]
∗∗ Postal address: School of Mathematics, University of Sussex, Falmer, Brighton, UK.
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Abstract

If Z is the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ZlogZ] < ∞, then there is a branching measure μ defined on ∂Γ, the set of all paths ξ which have a unique node ξ|n at each generation n. We use the natural metric ρ(ξ,η) = e−n, where n = max{k : ξ|k = η|k}, and observe that the local dimension index is d(μ,ξ) = limn→∞ log(μB(ξ|n))/(-n) = α = logm, for μ-almost every ξ. Our objective is to consider the exceptional points where the above display may fail. There is a nontrivial ‘thin’ spectrum for ̄d(μ,ξ) when p1 = P{Z = 1} > 0 and Z has finite moments of all positive orders. Because ̱d(μ,ξ) = a for all ξ, we obtain a ‘thick’ spectrum by introducing the ‘right’ power of a logarithm. In both cases, we find the Hausdorff dimension of the exceptional sets.

Keywords

Branching measureGalton-Watson treeHausdorff dimensionpacking dimensionmultifractal spectrum

MSC classification

Primary: 60G17: Sample path properties 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 100 - 111
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1]. Athreya, K. B., and Ney, P. (1972). Branching Processes. Springer, Berlin.Google Scholar
[2]. Bingham, N. H. (1988). On the limit of a supercritical branching process. In A Celebration of Applied Probability (J. Appl. Prob. Spec. Vol. 25A), ed. Gani, J., Applied Probability Trust, Sheffield, pp. 215–228.Google Scholar
[3]. Dembo, A., Peres, Y., Rosen, J., and Zeitouni, O. (2000). Thick points for spatial Brownian motions: multifractal analysis of occupation measure. Ann. Prob. 28, 1–35.CrossRefGoogle Scholar
[4]. Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
[5]. Hawkes, J. (1981). Trees generated by a simple branching process. J. London Math. Soc. 24, 373–384.CrossRefGoogle Scholar
[6]. Hu, X. and Taylor, S. J. (1997). The multifractal structure of stable occupation measure. Stoch. Process. Appl. 66, 283–299.CrossRefGoogle Scholar
[7]. Khoshnevisan, D., Peres, Y., and Xiao, Y. (2000). Lim sup random fractals. Electron. J. Prob. 5, No. 4.CrossRefGoogle Scholar
[8]. Liu, Q. S. (1996). The exact Hausdorff dimension of a branching set. Prob. Theory Relat. Fields 104, 515–538.CrossRefGoogle Scholar
[9]. Liu, Q. S. (1996). The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In Trees, eds Chauvin, B. et al., Birkhäuser, Basel.Google Scholar
[10]. Liu, Q. S. (2001). Local dimension of the branching measure on a Galton–Watson tree. Ann. Inst. H. Poincaré Prob. Statist. 37, 195–222.CrossRefGoogle Scholar
[11]. Liu, Q. S., and Shieh, N. R. (1999). A uniform limit law for the branching measure on a Galton–Watson tree. Asian J. Math. 3, 381–386.CrossRefGoogle Scholar
[12]. Lyons, R., Pemantle, R., and Peres, Y. (1995). Ergodic theory on Galton–Watson trees. Ergodic Theory Dynamical Systems 15, 593–619.Google Scholar
[13]. Pemantle, R., and Peres, Y. (1995). Galton–Watson trees with the same mean have the same polar sets. Ann. Prob. 23, 1102–1124.CrossRefGoogle Scholar
[14]. Perkins, E. A., and Taylor, S. J. (1998). The multifractal structure of super-Brownian motion. Ann. Inst. H. Poincaré Prob. Statist. 34, 97–138.CrossRefGoogle Scholar
[15]. Shieh, N. R., and Taylor, S. J. (1998). Logarithmic multifractal spectrum of stable occupation measure. Stoch. Process. Appl. 75, 249–261.CrossRefGoogle Scholar