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Power-law-distributed level crossings define fractal behavior
Phys. Rev. E 76, 031134 – Published 27 September, 2007
DOI: https://doi.org/10.1103/PhysRevE.76.031134
Abstract
A relationship is established between the autocorrelation function of continuous Gaussian and non-Gaussian stochastic processes and the discrete process that describes their zero or level crossings. Random fractals occur when the distribution for the number of crossings is described by a class of Markov processes whose singlefold statistics are the discrete analog of the Lévy-stable continuous probability densities.
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References (22)
- P. Lévy, Theorie de l’Addition des Variables Aléatoires (Gaulthier-Villars, Paris, 1937).
- Lévy Flights and Related Topics in Physics, edited by M. Schlesinger, G. Zaslavsky, and U. Frisch (Springer, Berlin, 1995).
- G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930).
- J. Goodman, Statistical Optics (Wiley, New York, 1985).
- F. Black and M. S. Scholes, J. Polit. Econ. 81, 637 (1973).
- F. W. Stuetael and K. van Harn, Ann. Probab. 5, 893 (1979).
- K. I. Hopcraft, E. Jakeman, and J. O. Matthews, J. Phys. A 35, L745 (2002).
- S. O. Rice, in Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).
- H. E. Hurst, Trans. Am. Soc. Civ. Eng. 116, 770 (1951).
- B. B. Mandelbrot and J. W. van Ness, SIAM Rev. 2, 422 (1968); B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982).
- J. H. Van Vleck and D. Middleton, Proc. IEEE 54, 2 (1966).
- M. V. Berry, Philos. Trans. R. Soc. London, Ser. A 273, 611 (1973).
- E. Jakeman, Waves Random Media 2, 153 (1991); R. J. Adler and D. Firmin, Philos. Trans. R. Soc. London, Ser. A 303, 433 (1981); J. P. Graves, J. Horacek, R. A. Pitts, and K. I. Hopcraft, Plasma Phys. Controlled Fusion 47, L1 (2005).
- J. O. Matthews, K. I. Hopcraft, and E. Jakeman, J. Phys. A 36, 11585 (2003).
- Handbook of Mathematical Functions, 9th ed., edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1970).
- E. Renshaw, J. Appl. Probab. 25, 744 (1988); G. C. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070601 (2002).
- E. Jakeman, K. I. Hopcraft, and J. O. Matthews, J. Phys. A 38, 6447 (2005).
- E. Jakeman and K. D. Ridley, Modelling Fluctuations in Scattered Waves, (Taylor & Francis, New York, 2006).
- D. Blacknell, M. Jahangir, F. Piccolo, and R. J. A Tough, J. Phys. A 43, 1231 (2001).
- E. Jakeman, Opt. Acta 28, 435 (1981).
- J. M. Smith, K. I. Hopcraft, and E. Jakeman (unpublished).
- T. J. Shepherd, Opt. Acta 31, 1399 (1984); E. Jakeman, K. I. Hopcraft, and J. O. Matthews, Proc. R. Soc. London, Ser. A 459, 623 (2003).