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

The ordinary Levy motion (oLm) 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 pf this paper is to study the influence of these effects on the self-affine properties of the oLm, namely, on the '1/alpha laws', i.e., time-dependence of the gth order structure function and of the range. Analytical estimates and simulations of the Anile 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. (C) 2000 Elsevier Science Ltd. All rights reserved.