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We introduce a maximal inequality for a local empirical process understrongly mixing data. Local empirical processes are defined as the (local)averages $\frac{1}{nh}\sum_{i=1}^n \mathbf{1}\{x - h \leq X_i \leqx+h\}f(Z_i)$, where $f$ belongs to a class of functions, $x \in \mathbb{R}$ and$h > 0$ is a bandwidth. Our nonasymptotic bounds control estimation erroruniformly over the function class, evaluation point $x$ and bandwidth $h$. Theyare also general enough to accomodate function classes whose complexityincreases with $n$. As an application, we apply our bounds to function classesthat exhibit polynomial decay in their uniform covering numbers. Whenspecialized to the problem of kernel density estimation, our bounds revealthat, under weak dependence with exponential decay, these estimators achievethe same (up to a logarithmic factor) sharp uniform-in-bandwidth rates derivedin the iid setting by \cite{Einmahl2005}.

A maximal inequality for local empirical processes under weak dependence