Study of the non-linear autocorrelations within the Gaussian regime

In this work we extend the recently considered toy model of Weierstrass or Lévy walks with varying velocity of the walker [1] by introducing a more realistic possibility that the walk can be occasionally intermitted by its momentary localization; the localizations themselves are again described by the Weierstrass or Lévy process. The direct empirical motivation for developing this combined model is, for example, the dynamics of financial high-frequency time series or hydrological and even meteorological ones where variations of the index are randomly intermitted by flat intervals of different length exhibiting no changes in the activity of the system. This combined Weierstrass walks was developed in the framework of the non-separable generalized continuous-time random walk (GCTRW) formalism developed recently [2]. This approach makes it possible to study by stochastic simulations the whole spatial-temporal range while analytically we can study only the initial, pre-asymptotic and asymptotic regions (but not the intermediate one). Our approach is possible since the Weierstrass walks is a geometric superposition of regular random walks each of which can be simply treated by stochastic simulations. This non-Markovian two-state (walking-localization) model makes possible to cover by the unified treatment a broad band of known up to now types of non-biased diffusion from the dispersive one over the normal, enhanced, ballistic, and hyperdiffusion up to the Richardson law of diffusion which defines here a part of the borderline which separates the latter from the `Lévy ocean' where the total mean-square displacement of the walker diverges. We observed that anomalous diffusion is characterized here by three fractional exponents: temporal one characterizing the localized state and two, temporal and spatial ones, characterizing the walking state. By considering successive dynamic (even) exponents we constructed a series of different diffusion phase diagrams on the plane defined by the spatial and temporal fractional dimensions of the walking state. To adapt the model to the description of empirical data (or discrete time series) which are collected with a discrete time-step we used in the continuous-time series produced by the model a discretization procedure. We observed that such a procedure generates, in general, long-range non-linear autocorrelations even in the Gaussian regime, which appear to be similar to those observed, e.g., in the financial time series [3–6], although single steps of the walker within continuous time are, by definition, uncorrelated. This suggests a surprising explanation alternative to the one proposed very recently (cf. [7] and Refs. therein) although both approaches involve related variants of the well-known CTRW formalism applied yet in many different branches of knowledge [8–10]. Copyright EDP Sciences, Springer-Verlag 2003