Modeling long-range memory with stationary Markovian processes

TL;DR

This paper constructs explicit stationary Markovian processes with Gaussian or exponential tails that exhibit power-law long-range correlations, challenging the notion that long memory requires heavy-tailed distributions.

Contribution

It introduces a method to generate Markovian processes with long-range dependence and light tails, expanding modeling options for complex systems with memory.

Findings

Markovian processes with Gaussian/exponential tails can have power-law autocorrelation decay.

Long-range correlations are not necessarily linked to heavy-tailed distributions.

The processes can be described by Langevin equations, enabling continuous-time modeling of long memory.

Abstract

In this paper we give explicit examples of power-law correlated stationary Markovian processes y(t) where the stationary pdf shows tails which are gaussian or exponential. These processes are obtained by simply performing a coordinate transformation of a specific power-law correlated additive process x(t), already known in the literature, whose pdf shows power-law tails 1/x^a. We give analytical and numerical evidence that although the new processes (i) are Markovian and (ii) have gaussian or exponential tails their autocorrelation function still shows a power-law decay <y(t) y(t+T)>=1/T^b where b grows with a with a law which is compatible with b=a/2-c, where c is a numerical constant. When a<2(1+c) the process y(t), although Markovian, is long-range correlated. Our results help in clarifying that even in the context of 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 processes associated to Langevin equations thus showing that long-memory effects can be modeled in the context of continuous time stationary Markovian processes.