On non-parametric estimation of the Lévy kernel of Markov processes |
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Authors: | Florian A.J. Ueltzhö fer |
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Affiliation: | Lehrstuhl für mathematische Statistik, Technische Universität München, Boltzmannstraße 3, D-85 748 Garching b. M., Germany; TUM Institute for Advanced Study, Technische Universität München, Lichtenbergstraße 2a, D-85 748 Garching b. M., Germany |
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Abstract: | We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations X0,XΔ,…,XnΔ, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞ and Δ→0) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided. |
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Keywords: | primary, 62M05 secondary, 62G07, 60F05, 60J25 |
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