Warped Infinitely Divisible Cascades: Beyond Power Laws. Cascades Infiniment Divisibles Voilées: Au-Delà des Lois de Puissance

Warped Infinitely Divisible Cascades: Beyond Power Laws

Cascades Infiniment Divisibles Voilées: Au-Delà des Lois de Puissance

Pierre Chainais Rudolf Riedi  Patrice Abry 

ISIMA-LIMOS UMR 6158 – Université Clermont II,Aubière France

Depts. of Statistics and of ECE, Rice University, Houston Texas, USA

CNRS UMR 5672, Laboratoire de Physique. ENS Lyon, France

9 February 2004
28 February 2005
| Citation



We address the definitions and synthesis of stochastic processes which possess warped scaling laws that depart from power law behaviors in a controlled manner. We define warped infinitely divisible cascading (IDC) noise,motion and random walk. We provide a theoretical derivation of the scaling behavior of the moments of their increments. We provide numerical simulations of a warped log-Normal cascade to illustrate these results. Algorithms for synthesis and Matlab functions are available from our web pages. 


Nous présentons les définitions et synthèses de processus stochastiques respectant des lois d'échelles voilées, qui s'écartent de façon contrôlée d'un comportement en loi de puissance. Nous définissons des bruit, mouvement et marche aléatoire issus de cascades infiniment divisibles (IDC) voilées. Nous étudions analytiquement le comportement des moments des accroissements de ces processus à travers les échelles. Ces résultats théoriques sont illustrés sur l'exemple d'une cascade log-Normale voilée. Les algorithmes de synthèse et les fonctions Matlab utilisés sont disponibles sur nos pages web.


Fractional Brownian motion,infinitely divisible cascades,multifractal processes,multiplicative cascades, multiscaling,random walk,turbulence

Mots clés 

Mouvement Brownien fractionnaire,cascades infiniment divisibles,processus multifractals,cascades multiplicatives, multiscaling,marche aléatoire,turbulence.

1. Introduction
2. IDC Noise
3. IDC Motion & Random Walk
4. Scaling Behavior of IDC
5. Evolution of the Increments Distributions
6. Numerical Validations
7. Conclusion

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