Theory of Probability &Its Applications, Volume 66, Issue 3, Page 455-468, January 2021.
Let $\widehat F_n$ be the smooth empirical estimator obtained by integrating a kernel type density estimator based on a random sample of size $n$ from continuous distribution function $F$. The almost sure deviation between smooth empirical and smooth quantile processes is investigated under $\phi$-mixing and strong mixing conditions. We derive a pointwise as well as a uniform Bahadur--Kieffer type representation for smooth quantiles under cases of $\phi$-mixing and strong mixing. These results extend those of Babu and Singh [J. Multivariate Anal., 8 (1978), pp. 532--549] and Ralescu [J. Statist. Plann. Inference, 32 (1992), pp. 243--258].

中文翻译：

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相关随机变量的平滑经验和分位数过程之间的一些渐近性质

Theory of Probability &Its Applications，第 66 卷，第 3 期，第 455-468 页，2021 年 1 月。
令 $\widehat F_n$ 为平滑经验估计量，该估计量是通过对基于大小为 $n$ 的随机样本的核类型密度估计量积分而获得的连续分布函数 $F$。在$\phi$-混合和强混合条件下，研究了平滑经验和平滑分位数过程之间几乎肯定的偏差。在 $\phi$ 混合和强混合的情况下，我们为平滑分位数推导出逐点以及均匀的 Bahadur--Kieffer 类型表示。这些结果扩展了 Babu 和 Singh [J. Multivariate Anal., 8 (1978), pp. 532--549] 和 Ralescu [J. 国家主义者。计划。推论，32 (1992)，第 243--258 页]。