Tempered fractionally integrated process with stable noise as a transient anomalous diffusion model

Abstract

We present here the autoregressive tempered fractionally integrated moving average (ARTFIMA) process obtained by taking the tempered fractional difference operator of the non-Gaussian stable noise. The tempering parameter makes the ARTFIMA process stationary for a wider range of the memory parameter values than for the classical autoregressive fractionally integrated moving average, and leads to semi-long range dependence and transient anomalous behavior. We investigate ARTFIMA dependence structure with stable noise and construct Whittle estimators. We also introduce the stable Yaglom noise as a continuous version of the ARTFIMA model with stable noise. Finally, we illustrate the usefulness of the ARTFIMA process on a trajectory from the Golding and Cox experiment.