A large deviation theorem for U-processes

This paper develops a large deviation theorem for families of sample means of U-statistic structure (i.e., U-processes). These results extend the work of Sethuraman (1964) and Wu (1994) on large deviation theory for families of ordinary sample means and the classical empirical process. Along the way we obtain an extension to U-statistics of an important isoperimetric inequality of Talagrand (1994) for ordinary means. Applications include the simplicial depth function of Liu (1990) and sup-norm statistics (e.g., Kolmogorov-Smirnov type goodness-of-fit statistics) defined over U-processes.