The paper by Wassily Hoeffding introduces the concept of $U$-statistics, which are a class of statistics that are unbiased estimates of regular functionals of distribution functions (d.f.). A $U$-statistic is defined as $U = \sum' \Phi(X_{\alpha_1}, \cdots, X_{\alpha_m}) / (n(n-1)\cdots(n-m+1)$, where the sum is over all permutations of $m$ different integers from $1$ to $n$. If the $X_i$ are independent and have the same d.f. $F(x)$, $U$ is an unbiased estimate of the population characteristic $\theta(F) = \int \cdots \int \Phi(x_1, \cdots, x_m) dF(x_1) \cdots dF(x_m)$. Hoeffding establishes that if the $X_i$ have the same distribution and $\Phi(x_1, \cdots, x_m)$ is independent of $n$, the distribution of $\sqrt{n}(U - \theta)$ tends to a normal distribution as $n \to \infty$, under the condition that $E\{\Phi^2(X_1, \cdots, X_m)\}$ exists. Similar results are shown for joint distributions of multiple $U$-statistics, statistics asymptotically equivalent to $U$, and certain functions of $U$ or $U'$. The paper also discusses the asymptotic distribution of rank correlation statistics and their power functions for non-parametric tests of independence. The variance of a $U$-statistic is studied, and inequalities satisfied by the quantities $\xi_c$ are derived. The variance of $U$ is shown to be of order $n^{-1}$ if $\xi_1 > 0$, and of order $n^{-d-1}$ if the regular functional $\theta(F)$ is stationary of order $d$ for $F = F_0$. The paper also proves the asymptotic normality of $U$ under certain conditions, and extends these results to the case where the $X_i$ have different distributions.The paper by Wassily Hoeffding introduces the concept of $U$-statistics, which are a class of statistics that are unbiased estimates of regular functionals of distribution functions (d.f.). A $U$-statistic is defined as $U = \sum' \Phi(X_{\alpha_1}, \cdots, X_{\alpha_m}) / (n(n-1)\cdots(n-m+1)$, where the sum is over all permutations of $m$ different integers from $1$ to $n$. If the $X_i$ are independent and have the same d.f. $F(x)$, $U$ is an unbiased estimate of the population characteristic $\theta(F) = \int \cdots \int \Phi(x_1, \cdots, x_m) dF(x_1) \cdots dF(x_m)$. Hoeffding establishes that if the $X_i$ have the same distribution and $\Phi(x_1, \cdots, x_m)$ is independent of $n$, the distribution of $\sqrt{n}(U - \theta)$ tends to a normal distribution as $n \to \infty$, under the condition that $E\{\Phi^2(X_1, \cdots, X_m)\}$ exists. Similar results are shown for joint distributions of multiple $U$-statistics, statistics asymptotically equivalent to $U$, and certain functions of $U$ or $U'$. The paper also discusses the asymptotic distribution of rank correlation statistics and their power functions for non-parametric tests of independence. The variance of a $U$-statistic is studied, and inequalities satisfied by the quantities $\xi_c$ are derived. The variance of $U$ is shown to be of order $n^{-1}$ if $\xi_1 > 0$, and of order $n^{-d-1}$ if the regular functional $\theta(F)$ is stationary of order $d$ for $F = F_0$. The paper also proves the asymptotic normality of $U$ under certain conditions, and extends these results to the case where the $X_i$ have different distributions.