Functional limit theorems for the fractional Ornstein-Uhlenbeck process

TL;DR

This paper establishes a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, revealing joint Gaussian and non-Gaussian convergence, and introduces a rough creation result for non-Markovian processes.

Contribution

It provides the first joint convergence results for functionals of the fractional Ornstein-Uhlenbeck process with both Gaussian and non-Gaussian limits, applicable in various topologies.

Findings

Joint convergence to Gaussian and non-Gaussian limits

Validity for any L^2 functions and stronger integrability cases

Weak convergence of smooth curves to non-Markovian processes

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 H\"older 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.