Possible origin of the non-linear long-term 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 (Quantitative Finance 3 (2003) 201; Chem. Phys. 284 (2002) 481; Comp. Phys. Comm. 147 (2002) 565; Phys. A 264 (1999) 84; Phys. A 264 (1999) 107) 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 formalism developed very recently (Lecture Notes Comput. Sci. 2657 (2003) 407; Eur. Phys. J. B 33 (2003) 495). 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: one (temporal) characterizing the localized state and two (temporal and spatial) characterizing the walking one. By considering successive dynamic (even) exponents we constructed a series of different diffusion phase diagrams on the plane defined by the temporal and spatial (partial) fractional (dynamic) exponents characterizing the walking state. To adapt the model to the description of empirical data (the 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 (Phys. A 287 (2000) 396; Phys. A 299 (2001) 1; Phys. A 299 (2001) 16; Phys. A 299 (2001) 16), although single steps of the walker within continuous time are, by definition, uncorrelated. This suggests a suprising origin of long-range non-linear autocorrelations alternative to the one proposed very recently (cf. Mosaliver et al. (Phys. Rev. E 67 (2003) 021112) and refs. therein) although both approaches involve related variants of the well-known continuous-time random walk formalism applied yet in many different branches of knowledge (Phys. Rep. 158 (1987) 263; Phys. Rep. 195 (1990) 127; in: A. Bunde, S. Havlin (Eds.), Fractals in Science, Springer, Berlin, 1995, p. 1).