Superpositions of CARMA processes

Abstract

We introduce supCARMA processes, defined as superpositions of Lévy-driven CARMA processes with respect to a Lévy basis, as a natural extension of the superpositions of Ornstein-Uhlenbeck type processes. We then focus on supCAR(2) processes and show that they can be classified into three distinct types determined by the eigenstructure of the underlying CAR(2) matrix. For each type, we provide conditions for existence and derive explicit expressions for the correlation function. The resulting correlation structures may exhibit long-range dependence and can be non-monotone. These features make supCAR(2) processes a flexible class for modeling time series with oscillatory correlations or strong dependence.