Self - affinity of ordinary Levy motion, spurious multi - affinity and pseudo - Gaussian relations

Details Self - affinity of ordinary Levy motion, spurious multi - affinity and pseudo - Gaussian relations Self - affinity of ordinary Levy motion, spurious multi - affinity and pseudo - Gaussian relations

1999-07-16

arxiv.org/abs/cond-mat/9907234v1

Physics

Condensed Matter

Statistical Mechanics

13 pages, RevTeX 3.0, 7 figures PostScript

Scientific paper

The ordinary Levy motion is a random process whose stationary independent increments are statistically self-affine and distributed with a stable probability law characterized by the Levy index alpha, 0 < alpha < 2. The divergence of statistical moments of the order q > alpha leads to an important role of the finite sample effects. The objective of this paper is to study the influence of these effects on the self-affine properties of the ordinary Levy motion, namely, on the '1/alpha laws', that is, time dependence of the q-th order structure function and of the range. Analytical estimates and simulations of the finite sample effects clearly demonstrates three phenomena: spurious multi-affinity of the Levy motion, strong dependence of the structure function on the sample size at q > alpha, and pseudo-Gaussian behavior of the second-order structure function and of the normalized range. We discuss these phenomena in detail and propose the modified Hurst method for empirical rescaled range analysis.

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