Cramér type large deviations for some $U$-statistics

We prove Cramér type large deviations for some $U$-statistics of degree two with kernel $h(x,y)$ being of bounded variation on bounded rectangles. The proof consists of two basic steps. First some explicit bounds (similar to Helmers' bounds for $L$-statistics) for the $U$-statistics are obtained. Then Linnik's result and some results exploiting strong approximations are applied.