Advances in Applied Probability

Multifractal spectra for random self-similar measures via branching processes

J. D. Biggins, B. M. Hambly, and O. D. Jones

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Start with a compact set KRd. 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).

Article information

Source
Adv. in Appl. Probab., Volume 43, Number 1 (2011), 1-39.

Dates
First available in Project Euclid: 15 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aap/1300198510

Digital Object Identifier
doi:10.1239/aap/1300198510

Mathematical Reviews number (MathSciNet)
MR2761142

Zentralblatt MATH identifier
1223.28010

Subjects
Primary: 28A80: Fractals [See also 37Fxx] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G18: Self-similar processes

Keywords
Self-similar measure Hausdorff dimension general branching process multifractal spectrum local dimension

Citation

Biggins, J. D.; Hambly, B. M.; Jones, O. D. Multifractal spectra for random self-similar measures via branching processes. Adv. in Appl. Probab. 43 (2011), no. 1, 1--39. doi:10.1239/aap/1300198510. https://projecteuclid.org/euclid.aap/1300198510


Export citation

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.
  • Arbeiter, M. and Patzschke, N. (1996). Random self-similar multifractals. Math. Nachr. 181, 5–42.
  • Barral, J. (2000). Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Prob. 13, 1027–1060.
  • Barral, J. (2001). Generalized vector multiplicative cascades. Adv. Appl. Prob. 33, 874–895.
  • Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.
  • Biggins, J. D. (1979). Growth rates in the branching random walk. Z. Wahrscheinlichkeitsth. 48, 17–34.
  • Biggins, J. D. (1995). The growth and spread of the general branching random walk. Ann. Appl. Prob. 5, 1008–1024.
  • Biggins, J. D. (1998). Lindley-type equations in the branching random walk. Stoch. Process. Appl. 75, 105–133.
  • Biggins, J. D. and Kyprianou, A. E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Prob. 25, 337–360.
  • Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. Appl. Prob. 36, 544–581.
  • Chow, Y. S. and Teicher, H. (1978). Probability Theory. Springer, New York.
  • Falconer, K. (1990). Fractal Geometry. John Wiley, Chichester.
  • Falconer, K. J. (1994). The multifractal spectrum of statistically self-similar measures. J. Theoret. Prob. 7, 681–702.
  • Hambly, B. and Martin, J. B. (2007). Heavy tails in last-passage percolation. Prob. Theory Relat. Fields 137, 227–275.
  • Hutchinson, J. E. and Rüschendorf, L. (2000). Random fractals and probability metrics. Adv. Appl. Prob. 32, 925–947.
  • Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Process. Appl. 114, 27–50.
  • Jagers, P. (1975). Branching Processes with Biological Applications. Wiley-Interscience, London.
  • Jagers, P. (1989). General branching processes as Markov fields. Stoch. Process. Appl. 32, 183–212.
  • 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.
  • Latała, R. (1997). Estimation of moments of sums of independent real random variables. Ann. Prob. 25, 1502–1513.
  • Liang, J.-R. (2002). Random Markov-self-similar measures. Stoch. Process. Appl. 98, 113–130.
  • Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85–112.
  • Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286.
  • Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83–107.
  • 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.
  • 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.
  • Mauldin, R. D. and Williams, S. C. (1986). Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346.
  • Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.
  • Olsen, L. (1994). Random Geometrically Graph Directed Self-Similar Multifractals (Pitman Res. Notes Math. Ser. 307), Longman Scientific & Technical, Harlow.
  • Olsen, L. (1995). A multifractal formalism. Adv. Math. 116, 82–196.
  • Patzschke, N. (1997). The strong open set condition in the random case. Proc. Amer. Math. Soc. 125, 2119–2125.
  • 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.
  • Rockafellar, R. T. (1970). Convex Analysis (Princeton Math. Ser. 28). Princeton University Press.
  • Zähle, U. (1988). Self-similar random measures. I. Notion, carrying Hausdorff dimension, and hyperbolic distribution. Prob. Theory Relat. Fields 80, 79–100.