Universal Generation of Statistical Self-Similarity: A Randomized Central Limit Theorem

Abstract

A universal mechanism for the generation of statistical self-similarity-i.e., fractality in the context of random processes-is established. We consider a generic system which superimposes independent stochastic signals, producing a system output; all signals share a common statistical signal pattern, yet each signal has its own transmission parameters-amplitude, frequency, and initiation epoch. We characterize the class of parameter randomizations yielding statistically self-similar outputs in a universal fashion-i.e., for whatever signals fed into the system. Statistically self-similar outputs with finite variance further display (i) anomalous diffusion behavior-characterized by power-law temporal variance growth-and (ii) 1/f noise behavior-characterized by power-law power spectra. The mechanism presented is a "randomized central limit theorem" for fractal statistics of random processes.