We study deviation of U-statistics when samples have heavy-tailed distribution so the kernel of the U-statistic does not have bounded exponential moments at any positive point. We obtain an exponential upper bound for the tail of the U-statistics which clearly denotes two regions of tail decay, the first is a Gaussian decay and the second behaves like the tail of the kernel. For several common U-statistics, we also show the upper bound has the right rate of decay as well as sharp constants by obtaining rough logarithmic limits which in turn can be used to develop LDP for U-statistics. In spite of usual LDP results in the literature, processes we consider in this work have LDP speed slower than their sample size $n$.
翻译:我们研究U-统计学的偏差,因为样品的分布是重尾的,因此U-统计学的内核在任何正点上都没有被捆绑的指数性瞬间。我们获得了U-统计学的尾部的指数性上方线,它明显地代表两个尾部衰变区域,第一个是高斯衰变区域,第二个行为方式像内核的尾部。对于一些常见的U-统计学来说,我们还表明上界有正确的衰变速度以及尖锐的常数,通过获得粗略的对数限制,而后者又可用来开发U-统计学的LDP。尽管文献中通常有LDP结果,但我们认为这项工作的过程速度比其样本大小低。