The maximum of multifractal cascades: Exact distribution and approximations

D. Veneziano, and A. Langousis, The maximum of multifractal cascades: Exact distribution and approximations, Fractals, 13 (4), 311–324, 2005.



We study the distribution of the maximum M of multifractal measures using discrete cascade representations. For such discrete cascades, the exact distribution of M can be found numerically. We evaluate the sensitivity of the distribution of M to simplifying approximations, including independence of the measure among the cascade tiles and replacement of the dressing factor by a random variable with the same distribution type as the cascade generator. We also examine how the distribution of M varies with the dimensionality of the support and the multiplicity of the cascade. Of these factors, dependence of the measure among different cascade tiles has the highest effect on the distribution of M. This effect comes mainly from long-range dependence. We use these findings to propose a simple approximation to the distribution of M and give charts to implement the approximation for beta-lognormal cascades.

PDF Additional material: