  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 $n$.
