Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-25T06:19:18.777Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized vector multiplicative cascades

Part of: Stochastic processes Probability theory on algebraic and topological structures Classical measure theory Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Julien Barral
Julien Barral*
Affiliation:
INRIA Rocquencourt
*
∗ Postal address: Projet Fractales, INRIA Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France. Email address: julien.barral@inria.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We define the extension of the so-called ‘martingales in the branching random walk’ in R or C to some Banach algebras B of infinite dimension and give conditions for their convergence, almost surely and in the Lp norm. This abstract approach gives conditions for the simultaneous convergence of uncountable families of such martingales constructed simultaneously in C, the idea being to consider such a family as a function-valued martingale in a Banach algebra of functions. The approach is an alternative to those of Biggins (1989), (1992) and Barral (2000), and it applies to a class of families to which the previous approach did not. We also give a result on the continuity of these multiplicative processes. Our results extend to a varying environment version of the usual construction: instead of attaching i.i.d. copies of a given random vector to the nodes of the tree ∪n≥0N+n, the distribution of the vector depends on the node in the multiplicative cascade. In this context, when B=R and in the nonnegative case, we generalize the measure on the boundary of the tree usually related to the construction; then we evaluate the dimension of this nonstatistically self-similar measure. In the self-similar case, our convergence results make it possible to simultaneously define uncountable families of such measures, and then to estimate their dimension simultaneously.

Keywords

Self-similar cascadesbranching random walkrandom environmentrandom measuresHausdorff dimensionBanach-algebras-valued martingalesLp convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G57: Random measures 28A80: Fractals
Secondary: 60G42: Martingales with discrete parameter 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case) 60F25: $L^p$-limit theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 874 - 895
Copyright
Copyright © Applied Probability Trust 2001 

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

Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.Google Scholar
Barral, J. (1999). Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Prob. Theory Relat. Fields 113, 535–569.Google Scholar
Barral, J. (2000a). Differentiability of multiplicative processes related to branching random walks. Ann. Inst. H. Poincaré Prob. Statist. 36, 407–417.CrossRefGoogle Scholar
Barral, J. (2000b). Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Prob. 13, 1027–1060.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. (1989). Uniform convergence of martingales in the one-dimensional branching random walk. In Selected Proceedings of the Sheffield Symposium on Applied Probability (IMS Lecture Notes Monogr. Ser. 18), eds Basawa, I. V. and Taylor, R. L., Institute of Mathematical Statistics, Hayward, CA, pp. 159–173.Google Scholar
Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, 137–151.CrossRefGoogle Scholar
Billingsley, P. (1965). Ergodic Theory and Information. John Wiley, New York.Google Scholar
Brezis, H. (1983). Analyse Fonctionnelle, Théorie et Applications. Masson, Paris.Google Scholar
Čentsov, N., (1956). La convergence faible des processus stochastiques à trajectoires sans discontinuités de seconde espèce et l'approche dite ‘heuristique’ au tests du type de Kolmogorov–Smirnov. Teor. Veroyat. Primen. 1, 155–161 (in Russian with French summary). English translation: Weak convergence of stochastic processes whose trajectories have no discontinuity of the second kind and the ‘heuristic’ approach to the Kolmogorov–Smirnov tests. Theory Prob. Appl. 1, 140–144.Google Scholar
Collet, P. and Koukiou, F. (1992). Large deviations for multiplicative chaos. Commun. Math. Phys. 147, 329–342.CrossRefGoogle Scholar
Diestel, J. and Uhl, J. J. (1977). Vector Measures (Math. Surveys 15). American Mathematical Society, Providence, RI.Google Scholar
D'Souza, J. C. and Hambly, B. M. (1997). On the survival probability of a branching process in a random environment. Adv. Appl. Prob. 1, 38–55.Google Scholar
Durrett, R. and Liggett, T. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275–301.Google Scholar
Guivarc'h, Y., (1990). Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré Prob. Statist. 26, 261–285.Google Scholar
Holley, R. and Liggett, T. (1981). Generalized potlach and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165–195.Google Scholar
Joffe, A., Le Cam, L. and Neveu, J. (1973). Sur la loi des grands nombres pour les variables de Bernoulli attachées à un arbre dyadique. C. R. Acad. Sci. Paris A 277, 963–964.Google Scholar
Kahane, J.-P. (1987). Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. H. Poincaré Prob. Statist. 23, 289–296.Google Scholar
Kahane, J.-P. (1991). Produits de poids aléatoires et indépendants et applications. In Fractal Geometry and Analysis, eds Bélair, J. and Dubuc, S., Kluwer, Dordrecht, pp. 277–324.Google Scholar
Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoî t Mandelbrot. Adv. Math. 22, 131–145.Google Scholar
Kingman, J. F. C. (1975). The first birth problem for an age dependent branching process. Ann. Prob. 3, 790–801.Google Scholar
Koukiou, F. (1999). Directed polymers in random media and spin glass models on trees. In Classical and Modern Branching Processes (IMA Vols Math. Appl. 84), eds Athreya, K. B. and Jagers, P., Springer, Berlin, pp. 171–179.Google Scholar
Larsen, R. (1973). Banach Algebras—An Introduction (Pure Appl. Math. 24). Marcel Dekker, New York.Google Scholar
Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.Google Scholar
Liu, Q. S. (1996). On the survival probability of a branching process in a random environment. Ann. Inst. H. Poincaré Prob. Statist. 32, 1–10.Google Scholar
Liu, Q. S. (1997). Sur une équation fonctionnelle et ses application: une extension du théorème de Kesten–Stigum concernant des processus de branchements. Adv. Appl. Prob. 29, 353–373.Google Scholar
Liu, Q. S. (1998). Fixed points of a generalized smoothing transformation and applications to branching processes. Adv. Appl. Prob. 30, 85–112.Google Scholar
Liu, Q. S. (1999). Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks. Stoch. Process. Appl. 82, 61–87.Google Scholar
Liu, Q. S. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286.Google Scholar
Liu, Q. S. and Rouault, A. (1996). On two measures defined on the boundary of a branching tree. In Classical and Modern Branching Processes (IMA Vols Math. Appl. 84), eds Athreya, K. B. and Jagers, P., Springer, Berlin, pp. 187–201.Google Scholar
Lyons, R. (1996). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (IMA Vols Math. Appl. 84), eds Athreya, K. B. and Jagers, P., Springer, Berlin, pp. 217–221.Google Scholar
Mandelbrot, B. B. (1974a). Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358.Google Scholar
Mandelbrot, B. B. (1974b). Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées. C. R. Acad. Sci. Paris A 278, 289–292, 355–358.Google Scholar
Mandelbrot, B. B., Calvet, L. and Fisher, A. (1997). The multifractal model of asset return. Discussion Paper 1164, Cowles Foundation, Yale University. Available at http://cowles.econ.yale.edu/.Google Scholar
Molchan, G. M. (1996). Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179, 681–702.Google Scholar
Peyrière, J., (1977). Calculs de dimensions de Hausdorff. Duke Math. J. 44, 591–601.Google Scholar
Von Bahr, B. and Esseen, C.-G. (1965). Inequalities for the rth absolute moment of a sum of random variables 1≤ r≤ 2. Ann. Math. Statist. 36, 299–303.Google Scholar
Waymire, E. C. and Williams, S. C. (1995). Multiplicative cascades: dimension spectra and dependence. J. Fourier Anal. Applic., Spec. Issue 1995, 589–609.Google Scholar