Article ID Journal Published Year Pages File Type
1156717 Stochastic Processes and their Applications 2013 47 Pages PDF
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,…,XnΔX0,XΔ,…,XnΔ, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞nΔ→∞ and Δ→0Δ→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.

Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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