Asymptotic properties of autoregressive integrated moving average processes

In this paper we study the asymptotic behavior of so-called autoregressive integrated moving average processes. These processes constitute a large class of stochastic difference equations which includes among many other well-known processes the simple one-dimensional random walk. They were dubbed by G.E.P. Box and G.M. Jenkins who found them to provide useful models for studying and controlling the behavior of certain economic variables and various chemical processes. We show that autoregressive integrated moving average processes are asymptotically normally distributed, and that the sample paths of such processes satisfy a law of the iterated logarithm. We also establish a law which determines the time spent by a sample path on one or the other side of the "trend line" of the process.