Phillips–solo Asymptotics for Continuous-Time Arma Processes Sampled in Discrete Time

Phillips and Solo (1992, Annals of Statistics 20, 971–2001) provided sets of conditions concerning the summability of coefficients and the properties of innovations in discrete-time linear processes such that normalized sums satisfy certain asymptotic properties, such as laws of large numbers, central limit theorems, and invariance principles. In this article, we consider ARMA processes in continuous time that are sampled at discrete-time intervals and provide conditions on the continuous-time process such that the results of Phillips and Solo can be utilized for normalized sums of the discrete-time observations. We utilize a state-space representation in the case of stock and flow variables, both of which are a linear combination of the state vector, and derive the relevant asymptotics based on multivariate extensions of the Phillips–Solo results. An illustrative application of the results to unit (zero) root testing is provided.