Lévy flights, autocorrelation, and slow convergence

Previously we have put forward that the sluggish convergence of truncated Lévy flights to a Gaussian (Phys. Rev. Lett. 73 (1994) 2946) together with the scaling power laws in their probability of return to the origin (Nature 376 (1995) 46) can be explained by autocorrelation in data (Physica A 323 (2003) 601; Phys. Lett. A 315 (2003) 51). A purpose of this paper is to improve and enlarge the scope of such a result. The role of the autocorrelations in the convergence process as well as the problem of establishing the distance of a given distribution to the Gaussian are analyzed in greater detail. We show that whereas power laws in the second moment can still be explained by linear correlation of pairs, sluggish convergence can now emerge from nonlinear autocorrelations. Our approach is exemplified with data from the British pound–US dollar exchange rate.