Warped infinitely divisible cascades: beyond power laws

We address the definition 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.

[1]  A. Kolmogorov.  A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number , 1962, Journal of Fluid Mechanics.

[2]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[3]  D. Schertzer,et al.  Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes , 1987 .

[4]  Y. Gagne,et al.  Velocity probability density functions of high Reynolds number turbulence , 1990 .

[5]  E. Bacry,et al.  The Multifractal Formalism Revisited with Wavelets , 1994 .

[6]  Emmanuel Bacry,et al.  Random cascades on wavelet dyadic trees , 1998 .

[7]  B. Mandelbrot.  A Multifractal Walk down Wall Street , 1999 .

[8]  D. Veitch,et al.  Infinitely divisible cascade analysis of network traffic data , 2000, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100).

[9]  F. Schmitt,et al.  Stochastic equations generating continuous multiplicative cascades , 2001, cond-mat/0102346.

[10]  Pierre Chainais.  Cascades log-infiniment divisibles et anlayse multirésolution : application à l'étude des intermittences en turbulences , 2001 .

[11]  B. Mandelbrot,et al.  Multifractal products of cylindrical pulses , 2002 .

[12]  Jean Delour.  Processus aléatoire auto-similaires : applications en turbulence et en finance , 2001 .

[13]  E. Bacry,et al.  Multifractal random walk. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  E. Bacry,et al.  Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  R. Riedi,et al.  Multifractal products of stochastic processes: construction and some basic properties , 2002, Advances in Applied Probability.

[16]  E. Bacry,et al.  Log-Infinitely Divisible Multifractal Processes , 2002, cond-mat/0207094.

[17]  Murad S. Taqqu,et al.  Theory and applications of long-range dependence , 2003 .

[18]  B. Mandelbrot.  Intermittent turbulence in self-similar cascades : divergence of high moments and dimension of the carrier , 2004 .

[19]  P. Abry,et al.  SCALE INVARIANT INFINITELY DIVISIBLE CASCADES , 2004 .

[20]  Patrice Abry,et al.  On non-scale-invariant infinitely divisible cascades , 2005, IEEE Transactions on Information Theory.