On Non-parametric Estimation of the L\'evy Kernel of Markov Processes

TL;DR

This paper introduces a non-parametric estimator for the Lévy kernel density of recurrent Itô semi-martingale Markov processes, proving its consistency, asymptotic normality, and optimal convergence rates based on discrete observations.

Contribution

It develops a novel non-parametric estimation method for the Lévy kernel density of Markov processes, with proven statistical properties and asymptotic confidence intervals.

Findings

Estimator is consistent as sample size increases and observation interval decreases.

In positive recurrent cases, estimator is asymptotically normal.

In null recurrent cases, estimator is asymptotically mixed normal.

Abstract

We consider a recurrent Markov process which is an It\^o semi-martingale. The L\'evy kernel describes the law of its jumps. Based on observations X(0),X({\Delta}),...,X(n{\Delta}), we construct an estimator for the L\'evy kernel's density. We prove its consistency (as n{\Delta}->\infty and {\Delta}->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.