Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process

Abstract

An analytic expression for characteristic function defining a truncated L\'evy flight is derived. It is shown that the characteristic function yields results in agreement with recent simulations of truncated L\'evy flights by Mantegna and Stanley [Phys. Rev. Lett. 73, 2946 (1994)]. With the analytic expression for the characteristic function, the convergence of the L\'evy process towards the Gaussian is demonstrated without simulations. In the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures.