Hostname: page-component-7f64f4797f-7vssv Total loading time: 0 Render date: 2025-11-04T11:08:38.271Z Has data issue: false hasContentIssue false
  • English
  • Français

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: shiehnr@math.ntu.edu.tw
∗∗ Postal address: School of Mathematics, University of Sussex, Falmer, Brighton, UK.
Get access Rights & Permissions [Opens in a new window]

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 p 1 = 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.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 100 - 111
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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