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Variable Step Random Walks and Self-Similar Distributions

Gemunu H. Gunaratne·Joseph L. McCauley·Matthew Nicol·Andrei Torok
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Abstract

We study a scenario under which variable step random walks give anomalous statistics. We begin by analyzing the Martingale Central Limit Theorem to find a sufficient condition for the limit distribution to be non-Gaussian. We note that the theorem implies that the scaling index $ζ$ is 1/2. For corresponding continuous time processes, it is shown that the probability density function $W(x;t)$ satisfies the Fokker-Planck equation. Possible forms for the diffusion coefficient are given, and related to $W(x,t)$. Finally, we show how a time-series can be used to distinguish between these variable diffusion processes and Lévy dynamics.

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Gemunu H. Gunaratne, Joseph L. McCauley, Matthew Nicol, Andrei Torok. 2004-12-31. Variable Step Random Walks and Self-Similar Distributions. https://doi.org/10.1007/s10955-005-5474-y

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