Abstract: We present a stochastic superposition model which is capable of generating – in a universal fashion – various “fractal statistics”: anomalous diffusion, Levy laws, 1/f noises, and self-similarity. The model considers the superposition of independent stochastic processes: all processes sharing a common – yet arbitrary – stochastic process-pattern; each process having its own random parameters – initiation epoch, amplitude, and frequency. The stochastic superposition model is general and robust, and arises naturally in diverse fields of Science and Engineering: transmission channels and routers, and Internet servers in Communication; background noises in Physics and Electrical Engineering; probe dynamics in “stochastic baths”, and shot noise processes in Physics; river flows in Hydrology; tax revenues of states in Economics. Considering a specific process-statistic of the superposition model’s output, we focus on the following universality question: Is there a randomization of the processes’ parameters which renders the output’s process-statistic invariant with respect to the processes’ common stochastic pattern? The answer – for various process-statistics – turns out to be affirmative, and yields the aforementioned “fractal statistics”. This research thus establishes a unified framework that: (i) explains the universal emergence of “fractal statistics” in diverse fields of Science and Engineering, and (ii) provides an explicit model and randomization-algorithms that universally yield desired “fractal statistics”. The talk does not assume prior knowledge of stochastic processes.

Joint work with Joseph Klafter, Tel Aviv University.

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