Levy Flights, Autocorrelation, and Slow Convergence

Annibal Figueiredo, University of Brasilia
Iram Gleria, Federal University of Alagoas
Raul Matsushita, University of Brasilia
Sergio Da Silva, Federal University of Rio Grande Do Sul

Abstract

The sluggish convergence of truncated Lévy flights to a Gaussian together with the scaling power laws in their probability of return to the origin can be explained by autocorrelation in data. This paper enlarges the scope of such a result. The role of 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 pairwise correlation, sluggish convergence can now emerge from nonlinear autocorrelations. Our approach is exemplified with data from the British pound-US dollar exchange rate.