The European Physical Journal B

, Volume 66, Issue 1, pp 137–148 | Cite as

On discrete stochastic processes with long-lasting time dependence in the variance

Interdisciplinary Physics


In this manuscript, we analytically and numerically study statistical properties of an heteroskedastic process based on the celebrated ARCH generator of random variables whose variance is defined by a memory of qm-exponencial, form (eqm=1 x=ex). Specifically, we inspect the self-correlation function of squared random variables as well as the kurtosis. In addition, by numerical procedures, we infer the stationary probability density function of both of the heteroskedastic random variables and the variance, the multiscaling properties, the first-passage times distribution, and the dependence degree. Finally, we introduce an asymmetric variance version of the model that enables us to reproduce the so-called leverage effect in financial markets.


05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 89.90.+n Other topics in areas of applied and interdisciplinary physics 


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Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150Rio de JaneiroBrazil

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