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What is Learnable in Valiant's Theory of the Learnable? 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On Non-parametric Estimation of the Lévy Kernel of Markov Processes
Florian A. J. Ueltzhöfer · 2012-04-17 · via math.ST updates on arXiv.org

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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