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

翻译：我们针对强混合数据下的局部经验过程，引入一个极大不等式。局部经验过程定义为（局部）平均值 $\frac{1}{nh}\sum_{i=1}^n \mathbf{1}\{x - h \leq X_i \leq x+h\}f(Z_i)$，其中 $f$ 属于某个函数类，$x \in \mathbb{R}$，且 $h > 0$ 为带宽。我们的非渐近界一致地控制了在函数类、评估点 $x$ 和带宽 $h$ 上的估计误差。这些界具有足够的通用性，能够适应复杂度随 $n$ 增长而增加的函数类。作为应用，我们将这些界应用于均匀覆盖数呈多项式衰减的函数类。当专门应用于核密度估计问题时，我们的界揭示出：在具有指数衰减的弱依赖条件下，这些估计量能够达到与独立同分布情形下由 \cite{Einmahl2005} 推导出的、直至对数因子一致的锐化统一带宽速率。