Asymptotics of Time‐Varying Processes in Continuous‐Time Using Locally Stationary Approximations

We introduce a general theory on stationary approximations for locally stationary continuous‐time processes. Based on the stationary approximation, we use ‐weak dependence to establish laws of large numbers and central limit type results under different observation schemes. Hereditary properties for a large class of finite memory transformations show the flexibility of the developed theory. Sufficient conditions for the existence of stationary approximations for time‐varying Lévy‐driven state space models are derived and compared to existing results in Bitter, Stelzer, and Ströh (2023). We conclude with comprehensive results on the asymptotic behavior of the first and second‐order localized sample moments of time‐varying Lévy‐driven state space models.