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

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Abstract
An analytic expression for characteristic function defining a truncated Lévy flight is derived. It is shown that the characteristic function yields results in agreement with recent simulations of truncated Lévy flights by Mantegna and Stanley [Phys. Rev. Lett. 73, 2946 (1994)]. With the analytic expression for the characteristic function, the convergence of the Lévy 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.