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

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.

💡 Research Summary

The paper addresses a gap in the non‑parametric time‑series literature concerning uniform convergence of kernel‑based estimators when the design points are deterministic and equally spaced, i.e., (x_{t,T}=t/T) for (t=1,\dots,T). Existing results by Hansen (2008) and Kristensen (2009) rely on a random‑design framework where the design variable possesses a Lebesgue‑absolutely continuous density; such arguments break down for fixed grids that are ubiquitous in econometric time‑series applications.

The authors consider kernel averages of the form
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