Concentration of scalar ergodic diffusions and some statistical implications

TL;DR

This paper develops uniform concentration inequalities for scalar ergodic diffusions, providing a foundation for statistical analysis of diffusion processes and applications to density estimation errors.

Contribution

It introduces a systematic approach to concentration inequalities for diffusions using martingale approximation and generic chaining, extending tools from i.i.d. settings to continuous-time diffusions.

Findings

Established uniform concentration inequalities for ergodic diffusions.

Analyzed sup-norm error bounds for invariant density estimators.

Provided a probabilistic framework applicable to various Markov processes.

Abstract

We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. In the classical i.i.d. context, a key device for the statistical sup-norm analysis is provided by Talagrand-type concentration inequalities. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentration inequalities via martingale approximation and moment bounds obtained by the generic chaining method. The developed machinery is of independent probabilistic interest and can serve as a starting point for investigations of other processes such as more general Markov processes, in particular multivariate or discretely observed diffusions. As a first concrete statistical application, we analyse the sup-norm error of estimating the invariant density of an ergodic diffusion via the natural local time estimator and the classical nonparametric kernel density estimator, respectively.