Infinitely divisible time series models

In this paper we study time series models with infinitely divisible marginal distributions. The motivation for this study involved a project analyzing call center data. We study several different constructions of a class of autoregressive models and a class of moving average models with Gamma margins. The first one has a form similar to the classical time series ARMA model but with random coefficients. Maximum likelihood, conditional least squares and moment estimates are investigated. Their asymptotic properties are studied. The second construction comes from a general description of Gamma multivariate distributions. The third one generates continuous-time stationary Gamma processes based on the integration of Gamma random fields. All of them yield positively correlated Gamma processes. The last two constructions and most of the corresponding properties carry over to all infinitely divisible distributions with finite second moments. Constructions of higher order autoregressive models are also studied. The covariance structures and connections of those models are investigated. Open questions and future research directions are discussed. Related to these processes we describe a notation of decomposability for infinitely divisible distributions. It is shown that our constructions characterize all infinitely divisible, decomposable stationary processes