Autoregressive Sequences Via Lévy Processes

Abstract

We use L´evy processes to develop a family of first-order autoregressive sequences of random variables with values in R₊, called C-AR(1) processes. We obtain various distributional and regression properties for these processes and we establish a limit theorem that leads to the property of stationarity. A connection between stationarity of C-AR(1) processes and the notion of C-self-decomposability of van Harn and Steutel (1993) is discussed. A number of stationary C-AR(1) processes with specific marginals are presented and are shown to generalize several existing R₊-valued AR(1) models. The question of time reversibility is addressed and some examples are discussed.