Cram\'er type moderate deviation theorems for self-normalized processes

Cram\'er type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cram\'er type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.