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 α, 0<α<2. The divergence of statistical moments of the order q>α 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 oLm, namely, on the `1/α lawsʹ, i.e., time-dependence of the qth 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>α, 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.