Nonparametric inference on L\'evy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations

TL;DR

This paper introduces a spectral estimator for the Le9vy measure of a stationary Le9vy-driven Ornstein-Uhlenbeck process with compound Poisson subordinator, providing asymptotic distributions and methods for confidence band construction under macroscopic discrete observations.

Contribution

It proposes a novel spectral estimator for the Le9vy measure and establishes its asymptotic properties, including multivariate and high-dimensional CLTs, for processes observed at macroscopic intervals.

Findings

Derived multivariate central limit theorems for the estimator.

Established high-dimensional CLTs as the number of design points grows.

Developed practical methods for confidence band construction and bandwidth selection.

Abstract

This study examines a nonparametric inference on a stationary L\'evy-driven Ornstein-Uhlenbeck (OU) process $X = (X_{t})_{t \geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the L\'evy measure of the L\'evy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to construct confidence bands for the L\'evy measure and propose a practical method for bandwidth selection.