Asymptotics of time-varying processes in continuous-time using locally stationary approximations

TL;DR

This paper develops a comprehensive theory for analyzing continuous-time locally stationary processes using stationary approximations, establishing foundational laws and conditions for their asymptotic behavior.

Contribution

It introduces a general framework for stationary approximations in continuous-time processes, including new conditions for their existence and asymptotic properties.

Findings

Established laws of large numbers and CLT for locally stationary processes.

Derived conditions for stationary approximation existence in Lévydriven models.

Provided asymptotic results for localized sample moments.

Abstract

We introduce a general theory on stationary approximations for locally stationary continuous-time processes. Based on the stationary approximation, we use $\theta$-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 and infinite memory transformations show the flexibility of the developed theory. Sufficient conditions for the existence of stationary approximations for time-varying L\'evy-driven state space models are derived and compared to existing results. We conclude with comprehensive results on the asymptotic behavior of the first and second order localized sample moments of time-varying L\'evy-driven state space models.