On the asymptotic normality of kernel density estimators for linear random fields

TL;DR

This paper proves the asymptotic normality of kernel density estimators for causal linear random fields under weaker coefficient conditions, using the $m$-approximation method and a new CLT for $m$-dependent fields.

Contribution

It introduces weaker coefficient conditions for asymptotic normality of kernel density estimators in linear random fields and develops a novel CLT for unbounded $m$-dependent arrays.

Findings

Established asymptotic normality under weaker conditions

Proved a CLT for unbounded $m$-dependent random fields

Applied a recent moment inequality for stationary fields

Abstract

We establish sufficient conditions for the asymptotic normality of kernel density estimators, applied to causal linear random fields. Our conditions on the coefficients of linear random fields are weaker than known results, although our assumption on the bandwidth is not minimal. The proof is based on the $m$-approximation method. As a key step, we prove a central limit theorem for triangular arrays of stationary $m$-dependent random fields with unbounded $m$. We also apply a moment inequality recently established for stationary random fields.