Modeling long-range memory with stationary Markovian processes

Abstract

In this paper we give explicit examples of long-range correlated stationary Markovian processes y(t) where the stationary probability density function (pdf) shows tails which are Gaussian or exponential. These processes are obtained by simply performing appropriate coordinate transformations of a specific power-law correlated additive process x(t), already known in the literature, whose pdf shows power-law tails. We give analytical and numerical evidences that although the new processes are Markovian and have Gaussian or exponential tails, their autocorrelation function shows a power-law decay with logarithmic corrections. For a generic continuous and monotonously increasing coordinate transformation, we also analytically investigate what is the relationship between the asymptotic decay of the autocorrelation function and the tails of the stationary pdf. Extreme events seem to be associated to long-range correlated processes with power-law decaying autocorrelation function. However, the occurrence of extreme events is not necessary in order to have more general long-range correlated processes in which the autocorrelation shows a slow decay characterized by a power-law times a correction function such as the logarithm. Our results help in clarifying that even in the context of stationary Markovian processes long-range dependencies are not necessarily associated to the occurrence of extreme events. Moreover, our results can be relevant in the modeling of complex systems with long memory. In fact, we provide simple stationary processes associated to Langevin equations with white noise thus confirming that long-memory effects can be modeled in the context of continuous time stationary Markovian processes.