Estimation for stochastic damping Hamiltonian systems under partial observation. III. Diffusion term

TL;DR

This paper extends previous work on ergodic Hamiltonian systems by establishing a central limit theorem for a nonparametric estimator of the diffusion term under partial observation, completing the analysis of system parameters.

Contribution

It introduces a central limit theorem for a nonparametric estimator of the diffusion term in ergodic Hamiltonian systems with partial observation, building on prior results for density and drift estimation.

Findings

Central limit theorem for diffusion term estimator

Extension of previous results to diffusion parameter

Validation under partial observation conditions

Abstract

This paper is the third part of our study started with Cattiaux, Le\'{o}n and Prieur [Stochastic Process. Appl. 124 (2014) 1236-1260; ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384]. For some ergodic Hamiltonian systems, we obtained a central limit theorem for a nonparametric estimator of the invariant density [Stochastic Process. Appl. 124 (2014) 1236-1260] and of the drift term [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384], under partial observation (only the positions are observed). Here, we obtain similarly a central limit theorem for a nonparametric estimator of the diffusion term.