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Estimation of the Lévy measure: statistical inverse problem and Donsker Theorem

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We consider estimation of the characteristics of a Lévy process from discrete time observations. When the distance in observation time does not tend to zero, we face a nonlinear statistical inverse problem (that of decompounding in the compound Poisson case) with the Lévy measure as a nonparametric quantity. We shall concentrate on the generalized distribution function of the Lévy measure and show that a Donsker theorem or uniform CLT is feasible provided the characteristic functions does not decay too fast. Interestingly, for both, the pointwise CLT and the tightness part, we need advanced Fourier analysis (multiplier theorems, Besov spaces). Smoothed empirical processes, as introduced by Giné and Nickl (2008), will prove to be an essential ingredient. Joint work with Richard Nickl.

This talk is part of the Probability Theory and Statistics in High and Infinite Dimensions series.

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