Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T13:48:43.983Z Has data issue: false hasContentIssue false
  • English
  • Français

Multifractal spectra for random self-similar measures via branching processes

Part of: Stochastic processes Classical measure theory Markov processes

Published online by Cambridge University Press:  01 July 2016

J. D. Biggins ,
B. M. Hambly  and
O. D. Jones
J. D. Biggins*
Affiliation:
University of Sheffield
B. M. Hambly*
Affiliation:
University of Oxford
O. D. Jones*
Affiliation:
University of Melbourne
*
∗ Postal address: School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK. Email address: [email protected]
∗∗ Postal address: Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK.
∗∗∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Start with a compact set K ⊂ Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in Rd and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

Keywords

Self-similar measureHausdorff dimensiongeneral branching processmultifractal spectrumlocal dimension

MSC classification

Primary: 28A80: Fractals 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G18: Self-similar processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 1 - 39
Copyright
Copyright © Applied Probability Trust 2011 

References

Alsmeyer, G. and Iksanov, A. M. (2009). A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks. Electron J. Prob. 14, 289–312.Google Scholar
Arbeiter, M. and Patzschke, N. (1996). Random self-similar multifractals. Math. Nachr. 181, 5–42.Google Scholar
Barral, J. (2000). Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Prob. 13, 1027–1060.Google Scholar
Barral, J. (2001). Generalized vector multiplicative cascades. Adv. Appl. Prob. 33, 874–895.CrossRefGoogle Scholar
Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.Google Scholar
Biggins, J. D. (1979). Growth rates in the branching random walk. Z. Wahrscheinlichkeitsth. 48, 17–34.Google Scholar
Biggins, J. D. (1995). The growth and spread of the general branching random walk. Ann. Appl. Prob. 5, 1008–1024.CrossRefGoogle Scholar
Biggins, J. D. (1998). Lindley-type equations in the branching random walk. Stoch. Process. Appl. 75, 105–133.Google Scholar
Biggins, J. D. and Kyprianou, A. E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Prob. 25, 337–360.CrossRefGoogle Scholar
Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. Appl. Prob. 36, 544–581.CrossRefGoogle Scholar
Chow, Y. S. and Teicher, H. (1978). Probability Theory. Springer, New York.CrossRefGoogle Scholar
Falconer, K. (1990). Fractal Geometry. John Wiley, Chichester.Google Scholar
Falconer, K. J. (1994). The multifractal spectrum of statistically self-similar measures. J. Theoret. Prob. 7, 681–702.Google Scholar
Hambly, B. and Martin, J. B. (2007). Heavy tails in last-passage percolation. Prob. Theory Relat. Fields 137, 227–275.Google Scholar
Hutchinson, J. E. and Rüschendorf, L. (2000). Random fractals and probability metrics. Adv. Appl. Prob. 32, 925–947.CrossRefGoogle Scholar
Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Process. Appl. 114, 27–50.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. Wiley-Interscience, London.Google Scholar
Jagers, P. (1989). General branching processes as Markov fields. Stoch. Process. Appl. 32, 183–212.Google Scholar
Kolumbán, J., Soós, A. and Varga, I. (2003). Self-similar random fractal measures using contraction method in probabilistic metric spaces. Internat. J. Math. Math. Sci. 2003, 3299–3313.Google Scholar
Latała, R. (1997). Estimation of moments of sums of independent real random variables. Ann. Prob. 25, 1502–1513.Google Scholar
Liang, J.-R. (2002). Random Markov-self-similar measures. Stoch. Process. Appl. 98, 113–130.Google Scholar
Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85–112.Google Scholar
Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286.Google Scholar
Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83–107.Google Scholar
Liu, Q. and Rouault, A. (1997). On two measures defined on the boundary of a branching tree. In Classical and Modern Branching Processes (Minneapolis, MN, 1994; IMA Vol. Math. Appl. 84), Springer, New York, pp. 187–201.CrossRefGoogle Scholar
Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994; IMA Vol. Math. Appl. 84), Springer, New York, pp. 217–221.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
Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.Google Scholar
Olsen, L. (1994). Random Geometrically Graph Directed Self-Similar Multifractals (Pitman Res. Notes Math. Ser. 307), Longman Scientific & Technical, Harlow.Google Scholar
Olsen, L. (1995). A multifractal formalism. Adv. Math. 116, 82–196.Google Scholar
Patzschke, N. (1997). The strong open set condition in the random case. Proc. Amer. Math. Soc. 125, 2119–2125.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.Google Scholar
Rockafellar, R. T. (1970). Convex Analysis (Princeton Math. Ser. 28). Princeton University Press.Google Scholar
Zähle, U. (1988). Self-similar random measures. I. Notion, carrying Hausdorff dimension, and hyperbolic distribution. Prob. Theory Relat. Fields 80, 79–100.Google Scholar