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Angelika Rohde, Hamburg Print
Friday, 08 October 2010, 14:30 - 15:30

Uniform central limit theorems for multivariate diffusions

Angelika Rohde, University of Hamburg

Abstract: It has recently been shown that there are grave differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time. Besides establishing strong parallels to classical theory such as Ossiander's bracketing CLT and the general Gin'e-Zinn CLT for uniformly bounded families of functions, we find increased regularity also for multivariate ergodic diffusions, assuming that the invariant measure is finite with Lebesgue density $pi$. The effect is diminishing for growing dimension but always present. The fine differences to the classical iid setting are worked out using exponential inequalities for martingales and additive functionals of continuous Markov processes as well as the characterization of the sample path behavior of Gaussian processes by means of the generic chaining bound. To uncover the phenomenon, we study a smoothed version of the empirical diffusion process. It turns out that uniform weak convergence of the smoothed empirical process under necessary and sufficient conditions takes place with the mean-squared optimal bandwidth choice for a large collection of function classes where the invariant density is assumed to belong to.
This is a joint work with Claudia Strauch.

Location: 2NO salle des profs - Plaine
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