Long-Term Memory and Chaos: a Note

Abstract

Is it possible to study jointly long-term memory processes and chaotic processes? In a first sight, this interrogation can appear somewhat surprising because of the properties showed by the two kinds of processes. Effectively, a long-term memory process, like an Arfima process, is a stochastic one, while a chaotic process is by definition a deterministic one. However, this question finds its origins in recent works of Peters (1991, 1994) setting number of relations between the two processes. On the one hand, Peters showed that long-term memory and chaos concepts can be linked by the mean of the fractal dimension: “ Fractal time series are characterized as long memory processes. They possess cycles and trends, and are the result of a nonlinear dynamic system, or deterministic chaos ”(Peters, 1991, p. 119). In this way, a long-term memory process would be a process whose attractor has a fractal dimension. We can thus think that the link between fractal dimension and chaos might be set by the mean of the concept of strange attractor. However, we will show that this relation is useless in order to detect the presence of chaotic dynamics. On the other hand, according to Peters, long-term memory can be equivalently detected by R/S analysis and Lyapunov exponents since there exist nonperiodic cycles for the two processes: “ The long memory effect in equity prices has now been confirmed by two separate types of nonlinear analysis. R/S analysis on monthly S& P 500 stock returns found a biased random walk with a memory length about four years. The Lyapunov exponent for monthly inflation-detrended S& P 500 prices found a 42-month cycle ”(Peters, 1991, p. 180).