The dependence structure of log-fractional stable noise with analogy to fractional Gaussian noise

Abstract

We examine the process log-fractional stable motion (log-FSM), which is an α-stable process with α ∈ (1, 2). Its tail probabilities decay like x−α as x → ∞, and hence it has a finite mean, but its variance is infinite. As a result, its dependence structure cannot be described by using correlations. Its increments, log-fractional noise (log-FSN), are stationary and so the dependence between any two points in time can be determined by a function of only the distance (lag) between them. Since log-FSN is a moving average and hence “mixing,” the dependence between the two time points decreases to zero as the lag tends to infinity. Using measures such as the codifference and the covariation, which can replace the covariance when the variance is infinite, we show that the decay is so slow that log-FSN (or, conventionally, log-FSM) displays long-range dependence. This is compared to the asymptotic dependence structure of fractional Gaussian noise (FGN), a befitting circumstance since log-FSN and FGN share a number of features.

Keywords

• α-stable distribution
• codifference
• covariation
• fractional brownian motion
• log-fractional stable noise
• self-similar process