Concentration of scalar ergodic diffusions and some statistical implications

Cathrine Aeckerle-Willems, Claudia Strauch

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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 local time estimator and the classical nonparametric kernel density estimator, respectively.

Original languageEnglish
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Pages (from-to)1857-1887
Number of pages31
Publication statusPublished - Nov 2021


  • Concentration of diffusions
  • Ergodic diffusion
  • Exponential inequalities
  • Local time estimator


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