Variable Step Random Walks and Self-Similar Distributions

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 study the case when the scaling index∼ζ is∼12. 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.