Large deviations of bootstrapped U -statistics

We develop large deviation results with Cramér’s series and the best possible remainder term for bootstrapped U -statistics with non-degenerate bounded kernels. The method of the proof is based on the contraction technique of Keener, Robinson and Weber [R.W. Keener, J. Robinson, N.C. Weber, Tail probability approximations for U -statistics, Statist. Probab. Lett. 37 (1) (1998) 59–65], which is a natural generalization of the classical conjugate distribution technique due to Cramér [H. Cramér, Sur un nouveau théoréme-limite de la theorie des probabilites, Actual. Sci. Indust. 736 (1938) 5–23].