Ultra-slow convergence to a Gaussian: The truncated Lévy flight

Abstract

We introduce a class of quasi-stable stochastic process, the truncated Levy Flight (TLF). A TLF is a stochastic process with finite variance. We show theoretically and numerically that the convergence of the sum of n independent TLF to a. Gaussian process is usually extremely slow. In fact a remarkably large value of n can be required to ensure the convergence to a Gaussian process. We also investigate the statistical properties of the S&P 500 (a financial index) and we show that they are qualitatively in agreement with the one of a TLF.