Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields

TL;DR

This paper establishes the asymptotic normality of the Parzen-Rosenblatt kernel density estimator for stationary strongly mixing random fields using a novel approach based on Lindeberg's method, simplifying conditions on bandwidth and mixing coefficients.

Contribution

It introduces a new proof technique for asymptotic normality that requires minimal bandwidth conditions and offers a simple criterion for mixing coefficients.

Findings

Proves asymptotic normality under minimal conditions

Uses Lindeberg's method instead of traditional techniques

Provides a simple criterion for strong mixing coefficients

Abstract

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt (1956) and Parzen (1962)) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg's method rather than on Bernstein's small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the (non-uniform) strong mixing coefficients which do not depend on the bandwith.