Revista Matemática Iberoamericana

Poissonian products of random weights: Uniform convergence and related measures

Julien Barral

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The random multiplicative measures on $\mathbb{R}$ introduced in Mandelbrot ([Mandelbrot 1996]) are a fundamental particular case of a larger class we deal with in this paper. An element $\mu$ of this class is the vague limit of a continuous time measure-valued martingale $\mu _{t}$, generated by multiplying i.i.d. non-negative random weights, the $(W_M)_{M\in S}$, attached to the points $M$ of a Poisson point process $S$, in the strip $H=\{(x,y)\in \mathbb{R}\times\mathbb{R}_+ ; 0 < y\leq 1\}$ of the upper half-plane. We are interested in giving estimates for the dimension of such a measure. Our results give these estimates almost surely for uncountable families $(\mu ^{\lambda})_{\lambda \in U}$ of such measures constructed simultaneously, when every measure $\mu^{\lambda}$ is obtained from a family of random weights $(W_M(\lambda))_{M\in S}$ and $W_M(\lambda)$ depends smoothly upon the parameter $\lambda\in U\subset\mathbb{R}$. This problem leads to study in several sense the convergence, for every $s\geq 0$, of the functions valued martingale $Z^{(s)}_t: \lambda \mapsto \mu_{t}^{\lambda }([0,s])$. The study includes the case of analytic versions of $Z^{(s)}_t(\lambda)$ where $\lambda\in\mathbb{C}^n$. The results make it possible to show in certain cases that the dimension of $\mu^{\lambda}$ depends smoothly upon the parameter. When the Poisson point process is statistically invariant by horizontal translations, this construction provides the new non-decreasing multifractal processes with stationary increments $s\mapsto \mu ([0,s])$ for which we derive limit theorems, with uniform versions when $\mu$ depends on $\lambda$.

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Rev. Mat. Iberoamericana, Volume 19, Number 3 (2003), 813-856.

First available in Project Euclid: 20 February 2004

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Primary: 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx] 28A80: Fractals [See also 37Fxx] 60G18: Self-similar processes 60G44: Martingales with continuous parameter 60G55: Point processes 60G57: Random measures

Poisson point processes Banach space valued martingales random measures Hausdorff dimension


Barral, Julien. Poissonian products of random weights: Uniform convergence and related measures. Rev. Mat. Iberoamericana 19 (2003), no. 3, 813--856.

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  • Barral, J.: Differentiability of multiplicative processes related to branching random walks. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 407-417.
  • Barral, J.: Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Probab. 13 (2000), no. 4, 1027-1060.
  • Barral, J.: Generalized vector multiplicative cascades. Adv. in Appl. Probab. 33 (2001), 874-895.
  • Barral, J. and Mandelbrot, B.B.: Multifractal product of cylindrical pulses. Probab. Theory Related Fields 124 (2002), no. 3, 409-430 (and also a different version in Cowles Foundation Discussion paper no. 1287,
  • Biggins, J.D.: Uniform convergence of martingales in the one-dimensional branching random walk. In Selected Proceedings of the Sheffield Symposium on Applied Probability (Sheffield, 1989), 159-173. IMS Lecture Notes Monogr. Ser. 18. Inst. Math. Statist., Hayward, CA, 1991.
  • Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 (1992), 137-151.
  • Billingsley, P.: Ergodic Theory and Information, Wiley, N. York, 1965.
  • Brezis, H.: Analyse fonctionnelle, Théorie et applications. Masson, Paris, 1983.
  • Dacunha-Castelle, D. and Duflo, M.: Probabilités et statistiques. Tome 2. Problèmes à temps mobile. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, 1983.
  • Diestel, J. and Uhl, J.J.: Vector measures. Mathematical Surveys 15. American Mathematical Society, Providence, R.I., 1977.
  • Fan, A.: Equivalence et orthogonalité des mesures aléatoires engendrées par martingales positives homogènes. Studia Math. 98 (1991), %no. 3, 249-266.
  • Fan, A.: How many intervals cover a point in Dvoretzky covering?. Israel J. Math 131 (2002), 157-184.
  • Joffe, A., Le Cam, L. and Neveu, J.: Sur la loi des grands nombres pour les variables aléatoires de Bernouilli attachées à un arbre dyadique. C. R. Acad. Sci. Paris 278 (1973), 963-964.
  • Kahane, J.-P.: Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 (1985), 105-150.
  • Kahane, J.-P.: Positive martingales and random measures. Chinese Ann. of Math. Ser B. 8 (1987), no. 1, 1-12.
  • Kahane, J.-P.: Produits de poids aléatoires et indépendants et applications. In Fractal geometry and analysis (Montreal, PQ, 1989), 277-324. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 346. Kluwer Acad. Publ., Dordrecht, 1991.
  • Kahane, J.-P. and Peyrière, J.: Sur certaines martingales de Benoît Mandelbrot. Advances in Math. 22 (1976), no. 2, 131-145.
  • Ledoux, M. and Talagrand, M.: Probability in Banach spaces. Springer-Verlag, 1991.
  • Liu, Q. and Rouault, A.: On two measures defined on the limit set of a Galton-Watson tree. In Classical and modern branching processes (Minneapolis, MN, 1994), 187-201. IMA Vol. Math. Appl. 84. Springer, New York, 1997.
  • Mandelbrot B. B.: Renewal sets and random cutouts. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 145-157.
  • Mandelbrot, B. B.: Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées. C. R. Acad. Sci. Paris 278 (1974), 355-358.
  • Mandelbrot B. B.: Multifractal products of pulses. Unpublished memorandum, 1996.
  • Mandelbrot, B. B., Fisher, A. and Calvet, L.: A multifractal model of asset returns. Cowles Foundation Discussion Paper No 1164, 1997.
  • Neveu J.: Martingales à temps discret. Masson, Paris, 1972.
  • Stein, E.M. and Zygmund, A.: Smoothness and differentiability of functions. Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 3-4, 1960/1961, 295-307.
  • von Bahr, B. and Esseen, C.-G.: Inequalities for the $r$th absolute moment of a sum of random variables $1\leq r\leq 2$. Ann. Math. Statist. 36 (1965), 299-303.