Variable Step Random Walks and Self-Similar Distributions

Details Variable Step Random Walks and Self-Similar Distributions Variable Step Random Walks and Self-Similar Distributions

2004-12-31

arxiv.org/abs/physics/0412182v1

Physics

Data Analysis, Statistics and Probability

13pages, 2 figures

Scientific paper

10.1007/s10955-005-5474-y

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 $\zeta$ 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 L\'evy dynamics.

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