Time changes and stationarity issues for extended scalar autoregressive models

A scalar discrete or continuous time process is reducible to stationarity (RWS) if its transform by some smooth time change is weakly stationary. Different issues linked to this notion are here investigated for autoregressive (AR) models. AR models are understood in a large sense and may have time-varying coefficients. In the continuous time case the innovation may be of the semi-martingale type-such as compound Poisson noise; in the discrete case, the noise may not be Gaussian. Necessary and sufficient conditions for scalar AR models to be RWS are investigated, with explicit formulas for the time changes. Stationarity reduction issues for discrete sequences sampled from time continuous AR processes are also considered. Several types of time changes, RWS processes and sequences are studied with examples and simulation, including the classical multiplicative stationary AR models.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper investigates different issues related to stationarity reduction in autoregressive models, including both continuous and discrete time cases. Necessary and sufficient conditions for autoregressive models to be weakly stationary are explored, with explicit formulas for the time changes. Furthermore, the issue of stationarity reduction for discrete sequences sampled from continuous time autoregressive processes is also considered.