This paper by Wassily Hoeffding discusses the asymptotic normality of U-statistics, which are a class of statistics derived from symmetric functions of random samples. A U-statistic is defined as the average of a symmetric function over all possible subsets of a given size from the sample. Hoeffding shows that under certain conditions, such as the existence of the second moment of the function, the distribution of a U-statistic converges to a normal distribution as the sample size increases. This result is extended to the joint distribution of multiple U-statistics and to other statistics that are asymptotically equivalent to U-statistics. The paper also explores the properties of regular functionals, which are functionals of the distribution function that can be estimated by U-statistics. It demonstrates that U-statistics are optimal unbiased estimators of regular functionals and provides conditions under which their asymptotic normality holds. The paper also discusses the variance of U-statistics and its behavior as the sample size increases. The results are applied to various examples, including moments, Fisher's k-statistics, and rank correlation statistics. The paper concludes with theorems on the asymptotic distribution of U-statistics and related statistics under different conditions, including when the underlying distributions are different.This paper by Wassily Hoeffding discusses the asymptotic normality of U-statistics, which are a class of statistics derived from symmetric functions of random samples. A U-statistic is defined as the average of a symmetric function over all possible subsets of a given size from the sample. Hoeffding shows that under certain conditions, such as the existence of the second moment of the function, the distribution of a U-statistic converges to a normal distribution as the sample size increases. This result is extended to the joint distribution of multiple U-statistics and to other statistics that are asymptotically equivalent to U-statistics. The paper also explores the properties of regular functionals, which are functionals of the distribution function that can be estimated by U-statistics. It demonstrates that U-statistics are optimal unbiased estimators of regular functionals and provides conditions under which their asymptotic normality holds. The paper also discusses the variance of U-statistics and its behavior as the sample size increases. The results are applied to various examples, including moments, Fisher's k-statistics, and rank correlation statistics. The paper concludes with theorems on the asymptotic distribution of U-statistics and related statistics under different conditions, including when the underlying distributions are different.