Hierarchical clustering-jump approach to analogs of renormalization-group transformations in fractal random processes

Abstract

A stochastic analog of renormalization-group transformations for fractal random processes is suggested. We start from a translationally invariant continuous-time and -space random walk. The jump events are grouped into hierarchical clusters. The process of jump clustering is described by means of a set of random aggregation probabilities selected from certain probability laws. The number of jumps from a cluster is a random variable. The renormalization consists in replacing the jump events from a cluster by a single renormalized (i.e., overall) jump. Explicit conditions for the occurrence of statistical fractal jump distributions are derived. If the probability distribution for the number of the jump events from a cluster is fractal, the same is true for the renormalized probability densities of the waiting time and displacement vector. The scaling of the jump events, space, and time is characterized by means of a unique fractal exponent. The renormalized stochastic process is compared with an alternative approach based on the use of random multiplicative processes. The comparison between the two formalisms illustrates L\'evy's theory of probability limit distributions with infinite moments. Although the rates of convergence and the corresponding parameters may be different, both models lead asymptotically to the same universal L\'evy distribution. The possible applications and generalizations of this method are also considered.