The Annals of Applied Probability

Convergence of complex multiplicative cascades

Julien Barral, Xiong Jin, and Benoît Mandelbrot

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Abstract

The familiar cascade measures are sequences of random positive measures obtained on [0, 1] via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 4 (2010), 1219-1252.

Dates
First available in Project Euclid: 20 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1279638785

Digital Object Identifier
doi:10.1214/09-AAP665

Mathematical Reviews number (MathSciNet)
MR2676938

Zentralblatt MATH identifier
1221.60028

Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles 60G18: Self-similar processes 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter
Secondary: 28A78: Hausdorff and packing measures

Keywords
Multiplicative cascades continuous function-valued martingales functional central limit theorem laws stable under random weighted mean multifractals

Citation

Barral, Julien; Jin, Xiong; Mandelbrot, Benoît. Convergence of complex multiplicative cascades. Ann. Appl. Probab. 20 (2010), no. 4, 1219--1252. doi:10.1214/09-AAP665. https://projecteuclid.org/euclid.aoap/1279638785


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References

  • [1] Arbeiter, M. and Patzschke, N. (1996). Random self-similar multifractals. Math. Nachr. 181 5–42.
  • [2] Arneodo, A., Bacry, E. and Muzy, J. F. (1998). Random cascades on wavelet dyadic trees. J. Math. Phys. 39 4142–4164.
  • [3] Bacry, E. and Muzy, J. F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 449–475.
  • [4] Bacry, E., Kozhemyak, A. and Muzy, J.-F. (2008). Continuous cascade models for asset returns. J. Econom. Dynam. Control 32 156–199.
  • [5] Barral, J. (1999). Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Related Fields 113 535–569.
  • [6] Barral, J. (2000). Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Probab. 13 1027–1060.
  • [7] Barral, J. (2001). Generalized vector multiplicative cascades. Adv. in Appl. Probab. 33 874–895.
  • [8] Barral, J. (2003). Poissonian products of random weights: Uniform convergence and related measures. Rev. Mat. Iberoamericana 19 813–856.
  • [9] Barral, J. and Jin, X. (2009). Multifractal analysis of complex random cascades. Comm. Math. Phys. To appear.
  • [10] Barral, J., Jin, X. and Mandelbrot, B. (2010). Uniform convergence for complex [0, 1]-martingales. Ann. Appl. Probab. 20 1205–1218.
  • [11] Barral, J. and Mandelbrot, B. B. (2002). Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 409–430.
  • [12] Barral, J. and Mandelbrot, B. B. (2004). Random multiplicative multifractal measures I, II, III. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot (M. Lapidus and M. V. Frankenkuijsen, eds.). Proc. Sympos. Pure Math. 72 3–90. Amer. Math. Soc., Providence, RI.
  • [13] Barral, J. and Mandelbrot, B. (2009). Fractional multiplicative processes. Ann. Inst. H. Poincaré Probab. Statist. 45 1116–1129.
  • [14] Barral, J. and Seuret, S. (2005). From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11 589–614.
  • [15] Barral, J. and Seuret, S. (2007). The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 214 437–468.
  • [16] Barral, J. and Seuret, S. (2007). Renewal of singularity sets of random self-similar measures. Adv. in Appl. Probab. 39 162–188.
  • [17] Bedford, T. (1989). Hölder exponents and box dimension for self-affine fractal functions. Constr. Approx. 5 33–48.
  • [18] Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 137–151.
  • [19] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [20] Breiman, L. (1992). Probability. Classics in Applied Mathematics 7. SIAM, Philadelphia, PA.
  • [21] Caliebe, A. and Rösler, U. (2003). Fixed points with finite variance of a smoothing transformation. Stochastic Process. Appl. 107 105–129.
  • [22] Chainais, P., Riedi, R. and Abry, P. (2005). On non-scale-invariant infinitely divisible cascades. IEEE Trans. Inform. Theory 51 1063–1083.
  • [23] Collet, P. and Koukiou, F. (1992). Large deviations for multiplicative chaos. Comm. Math. Phys. 147 329–342.
  • [24] Derrida, B., Evans, M. R. and Speer, E. R. (1993). Mean field theory of directed polymers with random complex weights. Comm. Math. Phys. 156 221–244.
  • [25] Durrett, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 275–301.
  • [26] Falconer, K. J. (1994). The multifractal spectrum of statistically self-similar measures. J. Theoret. Probab. 7 681–702.
  • [27] Fan, A. H. (2002). On Markov–Mandelbrot martingales. J. Math. Pures Appl. (9) 81 967–982.
  • [28] Frisch, U. and Parisi, G. (1985). Fully developed turbulence and intermittency in turbulence. Proc. International Summer School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (Proc. Int. Sch. Phys., “Enrico Fermi,” Course LXXXVIII) 84–88. North-Holland, Amsterdam.
  • [29] Gilbert, A. C., Willinger, W. and Feldmann, A. (1999). Scaling analysis of conservative cascades, with applications to network traffic. IEEE Trans. Inform. Theory 45 971–991.
  • [30] Guivarc’h, Y. (1990). Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré Probab. Statist. 26 261–285.
  • [31] Gupta, V. K. and Waymire, E. C. (1993). A statistical analysis of mesoscale rainfall as a random cascade. J. Appl. Meteor. 32 251–267.
  • [32] Heurteaux, Y. (2003). Weierstrass functions with random phases. Trans. Amer. Math. Soc. 355 3065–3077 (electronic).
  • [33] Holley, R. and Waymire, E. C. (1992). Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Probab. 2 819–845.
  • [34] Jaffard, S. (1998). Oscillation spaces: Properties and applications to fractal and multifractral functions. J. Math. Phys. 39 4129–4141.
  • [35] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 105–150.
  • [36] Kahane, J.-P. (1987). Positive martingales and random measures. Chinese Ann. Math. Ser. B 8 1–12.
  • [37] Kahane, J.-P. (1987). Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. H. Poincaré Probab. Statist. 23 289–296.
  • [38] Kahane, J.-P. (1991). Produits de poids aléatoires indépendants et applications. In Fractal Geometry and Analysis (Montreal, PQ, 1989). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 346 277–324. Kluwer, Dordrecht.
  • [39] Kahane, J. P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 131–145.
  • [40] Kôno, N. (1986). On self-affine functions. Japan J. Appl. Math. 3 259–269.
  • [41] Liu, Q. (1996). The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In Trees (Versailles, 1995). Progress in Probability 40 51–80. Birkhäuser, Basel.
  • [42] Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95 83–107.
  • [43] Liu, Q. (2002). An extension of a functional equation of Poincaré and Mandelbrot. Asian J. Math. 6 145–168.
  • [44] Liu, Q. and Rouault, A. (2000). Limit theorems for Mandelbrot’s multiplicative cascades. Ann. Appl. Probab. 10 218–239.
  • [45] Liu, Q., Rio, E. and Rouault, A. (2003). Limit theorems for multiplicative processes. J. Theoret. Probab. 16 971–1014.
  • [46] Ludeña, C. (2008). Lp-variations for multifractal fractional random walks. Ann. Appl. Probab. 18 1138–1163.
  • [47] Mandelbrot, B. (1972). Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and Turbulence. (M. Rosenblatt and C. Van Atta, eds.). Lecture Notes in Phys. 12 333–351. Springer, Berlin.
  • [48] Mandelbrot, B. (1974). Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées. C. R. Acad. Sci. Paris 278 289–292 and 355–358.
  • [49] Mandelbrot, B. (1974). Intermittent turbulence in self-similar cascades: Divergence of hight moments and dimension of the carrier. J. Fluid. Mech. 62 331–358.
  • [50] Mandelbrot, B. B. (1997). Fractals and Scaling in Finance. Springer, New York.
  • [51] Mandelbrot, B. B. and Jaffard, S. (1997). Peano–Pólya motions, when time is intrinsic or binomial (uniform or multifractal). Math. Intelligencer 19 21–26.
  • [52] Mannersalo, P., Norros, I. and Riedi, R. H. (2002). Multifractal products of stochastic processes: Construction and some basic properties. Adv. in Appl. Probab. 34 888–903.
  • [53] Meyer, M. and Stiedl, O. (2003). Self-affine fractal variability of human heartbeat interval dynamics in health and disease. Eur. J. Appl. Physiol. 90 305–316.
  • [54] Molchan, G. M. (1996). Scaling exponents and multifractal dimensions for independent random cascades. Comm. Math. Phys. 179 681–702.
  • [55] Ossiander, M. and Waymire, E. C. (2000). Statistical estimation for multiplicative cascades. Ann. Statist. 28 1533–1560.
  • [56] Riedi, R. H. (2003). Multifractal processes. In Theory and Applications of Long-Range Dependence 625–716. Birkhäuser, Boston, MA.
  • [57] Riedi, R. H. and Lévy-Véhel, J. (1997). TCP Traffic is multifractal: A numerical study. Techical report INRIA-RR-3129, INRIA.
  • [58] Seuret, S. (2009). On multifractality and time subordination for continuous functions. Adv. Math. 220 936–963.
  • [59] Stanley, H. E., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Ivanov, P. C. and Peng, C. K. (1999). Statistical physics and physiology: Monofractal and multifractal approaches. Physica A 270 309–324.
  • [60] Urbański, M. (1990). The probability distribution and Hausdorff dimension of self-affine functions. Probab. Theory Related Fields 84 377–391.
  • [61] Waymire, E. C. and Williams, S. C. (1995). Multiplicative cascades: Dimension spectra and dependence. In Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993) 589–609.
  • [62] Waymire, E. C. and Williams, S. C. (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348 585–632.