Taiwanese Journal of Mathematics

Limit Theorems for Multiplicative Cascades in a Random Environment

Shunli Hao

Full-text: Open access


Let $\zeta = (\zeta_{0},\zeta_{1},\ldots)$ be a sequence of independent and identically distributed random variables. For $r \geq 2$, let $\mu_r$ be Mandelbrot's (limit) measure of multiplicative cascades defined with positive weights indexed by nodes of a regular $r$-ary tree, and let $Z^{(r)}$ be the mass of $\mu_r$. We study asymptotic properties of $Z^{(r)}$ and the sequence of random measures $(\mu_r)_{r}$ as $r \to \infty$. We obtain some laws of large numbers and a central limit theorem. The results extend ones established by Liu and Rouault (2000) and by Liu, Rio and Rouault (2003).

Article information

Taiwanese J. Math., Volume 21, Number 4 (2017), 943-959.

Received: 26 August 2014
Revised: 14 February 2017
Accepted: 16 February 2017
First available in Project Euclid: 27 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

self-similar cascades Mandelbrot's martingales random environment law of large numbers central limit theorem large deviations


Hao, Shunli. Limit Theorems for Multiplicative Cascades in a Random Environment. Taiwanese J. Math. 21 (2017), no. 4, 943--959. doi:10.11650/tjm/5216. https://projecteuclid.org/euclid.twjm/1501120842

Export citation


  • J. Barral, Moments, continuité, et analyse multifractale des martingales de Mandelbrot, Probab. Theory Related Fields 113 (1999), no. 4, 535–569.
  • J. Barral, X. Jin and B. Mandelbrot, Uniform convergence for complex $[0,1]$-martingales, Ann. Appl. Probab. 20 (2010), no. 4, 1205–1218.
  • ––––, Convergence of complex multiplicative cascades, Ann. Appl. Probab. 20 (2010), no. 4, 1219–1252.
  • J. D. Biggins and A. E. Kyprianou, Measure change in multitype branching, Adv. in Appl. Probab. 36 (2004), no. 2, 544–581.
  • L. Breiman, Probability, Classics in Applied Mathematics 7, SIAM, Philaddelphia, PA, 1992.
  • P. Collet and F. Koukiou, Large deviations for multiplicative chaos, Comm. Math. Phys. 147 (1992), no. 2, 329–342.
  • R. Durrett and T. M. Liggett, Fixed points of the smoothing transformation, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 275–301.
  • Y. Guivarc'h, Sur une extension de la notion de loi semi-stable, Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), no. 2, 261–285.
  • R. Holley and E. C. Waymire, Multifractal dimensions and scaling exponents for strongly bounded random cascades, Ann. Appl. Probab. 2 (1992), no. 4, 819–845.
  • J. P. Kahane and J. Peyrière, Sur certaines martingales de Benoit Mandelbrot, Advances in Math. 22 (1976), no. 2, 131–145.
  • D. Kuhlbusch, On weighted branching processes in random environment, Stochastic Process. Appl. 109 (2004), no. 1, 113–144.
  • X. G. Liang, Propriétés asymptotiques des martingales de Mandelbrot et des marches aléatoires branchantes, Thèse (2010), Univ. Bretagne-Sud, Vannes.
  • Q. Liu, Sur une équation fonctionelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement, Adv. in Appl. Probab. 29 (1997), no. 2, 353–373.
  • ––––, Fixed points of a generalized smoothing transformation and applications to the branching random walk, Adv. in Appl. Probab. 30 (1998), no. 1, 85–112.
  • ––––, On generalized multiplicative cascades, Stochastic Process. Appl. 86 (2000), no. 2, 263–286.
  • ––––, Asymptotic properties and absolute continuity of laws stable by random weighted mean, Stochastic Process. Appl. 95 (2001), no. 1, 83–107.
  • Q. Liu, E. Rio and A. Rouault, Limit theorems for multiplicative processes, J. Theoret. Probab. 16 (2003), no. 4, 971–1014.
  • Q. Liu and A. Rouault, Limit theorems for Mandelbrot's multiplicative cascades, Ann. Appl. Probab. 10 (2000), no. 1, 218–239.
  • B. B. Mandelbrot, Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire, C. R. Acad. Sci. Paris Sér. A 278 (1974), 289–292.
  • ––––, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974), no. 2, 331–358.
  • M. Menshikov, D. Petritis and S. Popov, A note on matrix multiplicative cascades and bindweeds, Markov Process. Related Fields 11 (2005), no. 1, 37–54.
  • E. C. Waymire and S. C. Williams, A cascade decomposition theory with applications to Markov and exchangeable cascades, Trans. Amer. Math. Soc. 348 (1996), no. 2, 585–632.