Fractal random processes with iterated logarithmic tails: A generalization of the Shlesinger-Hughes stochastic renormalization approach.

Abstract

A probabilistic interpretation of the Shlesinger-Hughes stochastic renormalization approach as a cascade of random amplification events is used to describe the dynamics of multiple hierarchical processes. For a positive random variable, a cascade of stochastic amplifications leads to distributions with long tails described by inverse power laws. A ``hierarchy of hierarchies'' of amplification events is generated by assuming that the mean number of amplification events is also subject to stochastic amplification. The tails of the distributions generated by this procedure are much broader than the ones given by an inverse power law: they are inverse powers of the logarithm of the random variable. By considering a cascade of cascades (similar to a Russian doll), the tails of the corresponding distributions are given by inverse powers of the multiple iterated logarithm of the random variable. This type of asymptotic behavior is universal in the sense that it is independent of the details of the initial probability densities. The results are of interest for the description of very slow processes in frozen systems.