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arXiv · 1204.3765

On Non-parametric Estimation of the Lévy Kernel of Markov Processes

Florian A. J. Ueltzhöfer
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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 X(0),X(Δ),...,X(nΔ), we construct an estimator for the Lévy kernel's density. We prove its consistency (as nΔ->\infty 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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Florian A. J. Ueltzhöfer. 2013-05-12. On Non-parametric Estimation of the Lévy Kernel of Markov Processes. https://doi.org/10.1016/j.spa.2013.04.023

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