Large Deviations of U-Statistics. I

We develop large-deviation results with explicit order terms and Cramér's series for nondegenerate U-statistics of degree m under Cramér-type conditions on the kernel. The method of the proof is based on the contraction technique of Keener, Robinson, and Weber 15], which is a natural generalization of the classical method of Cramér 10]. Other techniques used in the proofs include truncation, decoupling inequalities, Borell's inequality for Rademacher chaos, and a partitioning method to bound the degenerate remainder term.