Abstract
The general intermittency expansion is developed for the probability distribution of the limit lognormal multifractal process introduced by Mandelbrot (in Rosenblatt M, Van Atta C (eds.) Statistical Models and Turbulence. Lecture Notes in Physics, vol. 12, p. 333. Springer, New York, 1972) and constructed explicitly by Bacry et al. (Phys Rev E 64:026103, 2001). The structure of expansion coefficients is shown to be determined solely by that of the Selberg integral. The coefficients are computed in terms of the values of the Riemann zeta function at positive integers. For application, an explicit formula for the negative integral moments of the process is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Bacry, E., Delour, J., Muzy, J.-F.: Multifractal random walk. Phys. Rev. E 64, 026103 (2001)
Bacry, E., Delour, J., Muzy, J.-F.: Modelling financial time series using multifractal random walks. Physica A 299, 84–92 (2001)
Bacry, E., Muzy, J.-F.: Log-infinitely divisible multifractal random walks. Commun. Math. Phys 236, 449–475 (2003)
Barral, J., Mandelbrot, B.B.: Multifractal products of cylindrical pulses. Prob. Theory Relat. Fields 124, 409–430 (2002)
Calvet, L., Fisher, A.: Multifractality in asset returns: theory and evidence. Rev. Econ. Stat. LXXXIV, 381–406 (2002)
Goldberger, A., Amaral, L., Hausdorff, J., Ivanov, P., Peng, C., Stanley, H.: Fractal dynamics in physiology: alterations with disease and aging. Proce. Nat. Acad. Sci. USA 99, 2466–2472 (2002)
Ivanov, P., Amaral, L., Goldberger, A., Havlin, S., Rosenblum, M., Struzik, Z., Stanley, H.: Multifractality in human heartbeat dynamics. Nature 399, 461–465 (1999)
Kahane, J.-P.: Sur le chaos multiplicatif. Ann. Sci. Math. Que. 9, 105–150 (1985)
Kahane, J.-P.: Positive martingales and random measures. Chi. Ann. Math. 8B, 1–12 (1987)
Kahane, J.-P.: Produits de poids aléatoires indépendants et applications. In: Belair, J., Dubuc, S. (eds) Fractal Geometry and Analysis, p. 277. Kluwer, Boston (1991)
Mandelbrot, B.B.: Possible refinement of the log-normal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In: Rosenblatt, M., Van Atta, C. (eds) Statistical Models and Turbulence. Lecture Notes in Physics vol 12, p. 333. Springer, New York (1972)
Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974)
Mandelbrot, B.B. et al.: Limit lognormal multifractal measures. In: Gotsman, E.A.(eds) Frontiers of Physics: Landau Memorial Conference, p. 309. Pergamon, New York (1990)
Meneveau, C., Sreenivasan, K.R.: The multifractal nature of the turbulent energy dissipation. J. Fluid Mech. 224, 429–484 (1991)
Muzy, J.-F., Bacry, E.: Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws. Phys. Rev. E 66, 056121 (2002)
Ostrovsky, D.: Limit lognormal multifractal as an exponential functional. J. Stat. Phys. 116, 1491–1520 (2004)
Ostrovsky, D.: Functional Feynman-Kac equations for limit lognormal multifractals. J. Stat. Phys. 127, 935–965 (2007)
Riordan, J.: Combinatorial Identities. Wiley, New York (1968)
Schertzer, D., Lovejoy, S.: Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades. J. Geophys. Res. 92, 9693–9721 (1987)
Schertzer, D., Lovejoy, S., Schmitt, F., Chigirinskaya, Y., Marsan, D.: Multifractal cascade dynamics and turbulent intermittency. Fractals 5, 427–471 (1997)
Schmitt, F.: A causal multifractal stochastic equation and its statistical properties. Eur. J. Phys. B 34, 85–98 (2003)
Schmitt, F., Schertzer, D., Lovejoy, S.: Multifractal analysis of foreign exchange data. Appl. Stoch. Models Data Anal. 15, 29–53 (1999)
Selberg, A.: Remarks on a multiple integral. Norske Mat. Tidsskr. 26, 71–78 (1944)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ostrovsky, D. Intermittency Expansions for Limit Lognormal Multifractals. Lett Math Phys 83, 265–280 (2008). https://doi.org/10.1007/s11005-008-0225-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-008-0225-z