Asymptotic properties of $U$-statistics

Abstract:

Let $r$ be a fixed positive integer. A $U$-statistic ${U_n}$ is an average of a symmetric measurable function of $r$ arguments over a random sample of size $n$. Such a statistic may be expressed as an average of independent and identically distributed random variables plus a remainder term. We develop a Kolmogorov-like inequality for this remainder term as well as examine some of its (a.s.) convergence properties. We then relate these properties to the $U$-statistic. In addition, the asymptotic normality of ${U_N}$, where $N$ is a positive integer-valued random variable, is established under certain conditions.