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

The ordinary Levy motion (oLm) is a random process whose stationary. indepe ndent increments are statistically self-affine and distributed with a stabl e probability law characterized by the Levy index alpha, 0 < alpha < 2. The divergence of statistical moments of the order q > alpha leads to an impor tant 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 o Lm, namely, on the '1/alpha laws', i.e., time-dependence of the gth order s tructure function and of the range. Analytical estimates and simulations of the Anile sample effects clearly demonstrates three phenomena: spurious mu lti-affinity of the Levy motion, strong dependence of the structure functio n on the sample size at q > alpha, and pseudo-Gaussian behavior of the seco nd-order structure function and of the normalized range. We discuss these p henomena in detail and propose the modified Hurst method for empirical resc aled range analysis. (C) 2000 Elsevier Science Ltd. All rights reserved.