Limit Theorems for $U$-Processes

TL;DR: The U-process theory as discussed by the authors is a collection of U-statistics over a family H of kernels h of m variables, based on a probability measure P on (S,S).

Abstract: A U-process is a collection of U-statistics. Concretely, a U-process over a family H of kernels h of m variables, based on a probability measure P on (S,S), is the collection \(\{ U_{n}^{{(m)}}(h,P):h \in \mathcal{H}\} \) of U-statistics. This chapter is devoted to the asymptotic theory of U-processes: we are interested in finding conditions on H and P ensuring that the law of large numbers, or the central limit theorem, or the law of the iterated logarithm, hold for \( U_n^{(m)} (h,P), \)uniformly in h ∈ H. The theory of empirical processes deals with the same questions for the case m = 1, and U-process theory is patterned after it. This is a relatively new subject, at least in the generality presented here (H being an arbitrary collection of kernels which are defined on a general measurable space). There is therefore a need to indicate its usefulness. To this effect, a section on applications is added at the end of the chapter (Section 5.5); there we illustrate the use of each of the main theorems in this chapter by deriving properties of certain multidimensional generalizations of the median, and in general, of M-estimators, and by studying estimators of the cumulative hazard and distribution functions of a random variable based on truncated data.