Functional limit theorems for the fractional Ornstein-Uhlenbeck process

Abstract

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the joint convergence to a limit with both Gaussian and non-Gaussian components. This is valid for any $L^2$ functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology. As an application we prove a `rough creation' result, i.e. the weak convergence of a family of random smooth curves to a non-Markovian random process with rough sample paths. This includes the second order problem and the kinetic fractional Brownian motion model.

To appear in the Journal of Theoretical Probability. arXiv admin note: text overlap with arXiv:1911.12600

References in corpus (3)

• Central limit theorems for sequences of multiple stochastic integrals
• Averaging dynamics driven by fractional Brownian motion
• Integrated functionals of normal and fractional processes

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• Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging
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• Statistical Inference for the Rough Homogenization Limit of Multiscale Fractional Ornstein-Uhlenbeck Processes
• Lévy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics