分位数过程增量的函数极限律及其应用
Functional limit laws for the increments of the quantile process; with applications
摘要：
基于大小为$n\to\infty$的随机样本，建立了赋范分位数过程增量的对数函数极限律。我们推广了Deheuvels和Mason（12）得到的一个极限定律，证明了他们的结果在带宽$h$上一致成立，仅限于在$[h'_n,h''_n]$上变化，其中$\{h'_n\}_{n\geq1}$和$\{h''_n\}_{n\geq 1}$是适当的非随机序列.我们对待的情况下，样本观测遵循可能的非均匀分布。作为我们定理的结果，我们提供了最近邻密度估计的一致极限律，其精神与Deheuvels和Mason（13）给出的核型估计的一致极限律相同。
英文摘要：
We establish a functional limit law of the logarithm for the increments of the normed quantile process based upon a random sample of size $n\to\infty$. We extend a limit law obtained by Deheuvels and Mason (12), showing that their results hold uniformly over the bandwidth $h$, restricted to vary in $[h'_n,h''_n]$, where $\{h'_n\}_{n\geq1}$ and $\{h''_n\}_{n\geq 1}$ are appropriate non-random sequences. We treat the case where the sample observations follow possibly non-uniform distributions. As a consequence of our theorems, we provide uniform limit laws for nearest-neighbor density estimators, in the spirit of those given by Deheuvels and Mason (13) for kernel-type estimators.
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