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November, 1992 Multifractal Dimensions and Scaling Exponents for Strongly Bounded Random Cascades
Richard Holley, Edward C. Waymire
Ann. Appl. Probab. 2(4): 819-845 (November, 1992). DOI: 10.1214/aoap/1177005577


The multifractal structure of a measure refers to some notion of dimension of the set which supports singularities of a given order $\alpha$ as a function of the parameter $\alpha$. Measures with a nontrivial multifractal structure are commonly referred to as multifractals. Multifractal measures are being studied both empirically and theoretically within the statistical theory of turbulence and in the study of strange attractors of certain dynamical systems. Conventional wisdom suggests that various definitions of the multifractal structure of random cascades exist and coincide. While this is rigorously known to be the case for certain deterministic cascade measures, the same is not true for random cascades. The purpose of this paper is to pursue this theory for a class of random cascades. In addition to providing a new role for the modified cumulant generating function (structure function) studied by Mandelbrot, Kahane and Peyriere, the results have implications for the theoretical interpretation of empirical data on turbulence and rainfall distributions.


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Richard Holley. Edward C. Waymire. "Multifractal Dimensions and Scaling Exponents for Strongly Bounded Random Cascades." Ann. Appl. Probab. 2 (4) 819 - 845, November, 1992.


Published: November, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0786.60064
MathSciNet: MR1189419
Digital Object Identifier: 10.1214/aoap/1177005577

Primary: 60G57
Secondary: 76F05

Keywords: dimension , multifractal , multiplicative cascade , tree

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.2 • No. 4 • November, 1992
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