标题： 局部经验过程在弱相依下的最大不等式 显示英文标题

标题： A maximal inequality for local empirical processes under weak dependence

摘要： 我们引入了在强混合数据下局部经验过程的最大不等式。 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$f$属于一个函数类，$x \in \mathbb{R}$和$h > 0$是带宽。 我们的非渐近界在函数类、评估点$x$和带宽$h$上统一控制估计误差。 它们也足够通用，可以容纳复杂度随着$n$增加的函数类。 作为应用，我们将这些界应用于其一致覆盖数呈现多项式衰减的函数类。 当专门应用于核密度估计问题时，我们的界限表明，在具有指数衰减的弱依赖条件下，这些估计量达到了与\cite{Einmahl2005}在独立同分布情况下推导出的相同（以对数因子为界）的均匀带宽最优速率。 显示英文摘要

摘要： 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}.