Abstract
This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scaling exponents of structure functions of the measure, τ(q), and (ii) multifractal dimensions are considered for cascades with a generator vector (w 1...w c) of the general type. These problems were previously studied for independent strongly bounded variablesw i: 0<a<w i≦c. Consequently, a broad class of models used in applications, including Kolmogorov's long-normal model in turbulence, long-stable “universal” cascades in atmospheric dynamics, has not been covered. Roughly speaking, problems (i), (ii) are here solved under the condition that the scaling exists; the τ-function is calculated for all arguments (previously this was done for positiveq) and a new effect emerges: the τ-function can generally involve discontinuities in the first derivative as well as in the second.
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Communicated by Ya.G. Sinai
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Molchan, G.M. Scaling exponents and multifractal dimensions for independent random cascades. Commun.Math. Phys. 179, 681–702 (1996). https://doi.org/10.1007/BF02100103
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DOI: https://doi.org/10.1007/BF02100103