Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T21:06:30.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Thick and thin points for random recursive fractals

Part of: Classical measure theory Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

B. M. Hambly  and
O. D. Jones
B. M. Hambly*
Affiliation:
University of Oxford
O. D. Jones*
Affiliation:
University of Southampton
*
∗ Postal address: Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK. Email address: hambly@maths.ox.ac.uk
∗∗ Postal address: Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider random recursive fractals and prove fine results about their local behaviour. We show that for a class of random recursive fractals the usual multifractal spectrum is trivial in that all points have the same local dimension. However, by examining the local behaviour of the measure at typical points in the set, we establish the size of fine fluctuations in the measure. The results are proved using a large deviation principle for a class of general branching processes which extends the known large deviation estimates for the supercritical Galton-Watson process.

Keywords

General branching processlarge deviationslocal dimension

MSC classification

Primary: 60G57: Random measures
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 28A80: Fractals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 251 - 277
Copyright
Copyright © Applied Probability Trust 2003 

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.)

References

Barral, J. (1999). Moments, continuité et analyse multifractals des martingales de Mandelbrot. Prob. Theory Relat. Fields 113, 535–569.Google Scholar
Biggins, J. D. and Bingham, N. H. (1993). Large deviations in the supercritical branching process. Adv. Appl. Prob. 25, 757–772.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.Google Scholar
Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2000). Thick points for spatial Brownian motion: multifractal analysis of occupation measure. Ann. Prob. 28, 1–35.Google Scholar
Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2000). Thin points for Brownian motion. Ann. Inst. H. Poincaré Prob. Statist. 36, 749–774.CrossRefGoogle Scholar
Doney, R. A. (1972). A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.Google Scholar
Doney, R. A. (1976). On single- and multi-type general age-dependent branching processes. J. Appl. Prob. 13, 239–246.Google Scholar
Falconer, K. J. (1986). Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559–582.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Graf., S. (1987). Statistically self-similar fractals. Prob. Theory Relat. Fields 74, 357–392.CrossRefGoogle Scholar
Graf, S., Mauldin, R. D. and Williams, S. C. (1988). The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc. 71, No. 381.Google Scholar
Hambly, B. M. and Kumagai, T. (2001). Fluctuation of the transition density for Brownian motion on random recursive Sierpiński gaskets. Stoch. Process. Appl. 92, 61–85.CrossRefGoogle Scholar
Hutchinson, J. E. and Rüschendorff, L. (2000). Random fractals and probability metrics Adv. Appl. Prob. 32, 925–947.CrossRefGoogle Scholar
Kolumbán, J. and Soós, A. (2001). Self-similar random fractal measure using contraction method in probabilistic metric spaces. Preprint.Google Scholar
Liang, J. R. (2002). Random Markov-self-similar measures. Stoch. Process. Appl. 98, 113–130.Google Scholar
Liu, Q. (1996). The exact Hausdorff dimension of a branching set. Prob. Theory Relat. Fields 104, 515–538.CrossRefGoogle Scholar
Liu., Q. (1996). The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In Trees (Progress Prob. 40), eds Chauvin, B., Cohen, S. and Rouault, A.. Birkhäuser, Basel, pp. 51–80.Google Scholar
Liu, Q. (2001). Local dimensions of the branching measure on a Galton–Watson tree. Ann. Inst. H. Poincaré Prob. Statist. 37, 195–222.Google Scholar
Mauldin, R. D. and Williams, S. C. (1986). Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346.Google Scholar
Mörters, P. and Shieh, N. R. (2002). Thick and thin points for branching measure on a Galton–Watson tree. Statist. Prob. Lett. 58, 13–22.CrossRefGoogle Scholar
Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.Google Scholar
Patzschke, N. and Zähle, U. (1990). Self-similar random measures. IV. The recursive construction model of Falconer, Graf and Mauldin and Williams. Math. Nachr. 149, 285–302.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
Zähle, U., (1988). Self-similar random measures. I. Notion, carrying Hausdorff dimension and hyperbolic distribution. 80, 79–100.Google Scholar