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The Annals of Probability 1986, Vol. 14, No. 4, 1347-1358 LIMIT THEOREMS FOR A TRIANGULAR SCHEME OF U-STATISTICS WITH APPLICATIONS TO INTER-POINT DISTANCES BY S. RAO JAMMALAMADAKA1 AND SVANTE J ANSON University of California, Santa Barbara and Uppsala University The asymptotic distribution of a “triangular” scheme of U-statistics is studied. Two limit theorems, applicable in different situations, are given. One theorem yields convergence to a normal distribution; the other includes Poisson limits and other limit laws. Applications to statistics based on small interpoint distances in a sample are given. 1. Introduction. There has been a renewed interest in the limit theory relating to U-statistics (Hoe‘ding, 1948) especially focussing on degenerate kernels and nonnormal limits. See, for instance, Rubin and Vitale (1980), Dynkin and Mandelbaum (1983), and Berman and Eagleson (1983). In Section 2 of this paper we study a “triangular” scheme of U-statistics and establish their asymptotic normality under suitable conditions. It is worth noting that one can obtain normal limit laws even with degenerate kernels. Other innitely divisible limit laws for the triangular scheme, including the Poisson limits, are examined in the next section. Applications to limit distributions of statistics based on interpoint distances are discussed in Section 4. These applications, which were the source of motivation for the results derived here, were suggested by the studies one of us made about statistics based on spacings. [See, for instance, Holst and Rao (1981).] They were also sparked by the recent papers of Bickel and Breiman (1983) and Onoyama et al. (1983). Theorems yielding asymptotic normality in this setting have also been proved by Weber (1983) using a martingale approach. His theorems yield the same conclusion as Theorem 2.1, but under di‘erent (nonequivalent) conditions. A few words about notation: We write “ —gd ” to denote convergence in distribution an “ —gc ” to denote complete convergence [cf. Loeve (1963), page 178]. We write Po(}\) to denote the Poisson distribution with mean A and Np(p., 2) to denote a p-dimensional normal distribution with mean vector p. and covariance matrix 2. 2. Asymptotic normality of a triangular scheme of U-statistics. Let X1, X2,... be a sequence of independently and identically distributed (i.i.d.) random variables. We will use X and Y to denote two independent random Received October 1984; revised April 1985. ‘Formerly J. S. Rao. AMS 1980 subject classications. Primary 60F05, 60E07; secondary 62E20. Key words and phrases. U-statistics, triangular scheme, multigraph, limit laws, innitely divisible distributions, inter-point distances. 1347 ‘T Institute of Mathematical Statistics is collaborating with J STOR to digitize, preserve, and extend access to ef‘%}73i The Annals ofProbabiIity. 5T0 R ® www.jstor.org

Limit Theorems for a Triangular Scheme of $U$-Statistics with Applications to Inter-Point Distances