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：本文研究当样本具有重尾分布时U统计量的偏差问题，此时U统计量的核函数在任意正点上都不具有有界指数矩。我们得到了U统计量尾部的指数上界，该上界清晰地区分出两个尾部衰减区域：第一个区域呈现高斯衰减，第二个区域的行为类似于核函数的尾部。对于几种常见的U统计量，我们还通过获得粗糙对数极限证明了该上界具有正确的衰减速率和精确常数，这进而可用于推导U统计量的大偏差原理。与文献中常见的大偏差结果不同，本文考虑的随机过程的大偏差速度慢于其样本量$n$。