Uniform convergence of kernel averages under fixed design with heterogeneous dependent data

TL;DR

This paper establishes uniform convergence rates for kernel averages on fixed, equally-spaced design points under dependent data, extending nonparametric regression analysis to deterministic time series data without stationarity assumptions.

Contribution

It provides the first uniform convergence rates for kernel averages on fixed grids with dependent, non-stationary data, using a novel grid-structure-based analysis.

Findings

Derived weak and strong uniform consistency rates under mixing conditions.

Extended the analysis to dependent triangular arrays.

Applied results to local linear estimators in nonparametric regression.

Abstract

We provide uniform convergence rates for kernel averages on $[0,1]$ under equally-spaced fixed design points of the form $x_{t,T}=t/T,\ t\in\{1,\dotsc, T\},\ T\in\mathbb{N}$. The rates of weak and strong uniform consistency are derived under strong mixing and moment conditions and do not require stationarity. The analysis exploits the grid structure and thus complements existing random-design results such as those of Hansen (2008) and Kristensen (2009), which rely on density-based conditioning arguments. The framework accommodates dependent triangular arrays and is particularly relevant for nonparametric methods applied to time series observed on deterministic grids. As an application, we derive uniform convergence rates for the local linear estimator in a nonparametric regression model with time-varying autoregressive errors. The theoretical results are illustrated through Monte Carlo experiments and an empirical application.