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

摘要: 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 California, Santa Barbara and Uppsala asymptotic distribution a “triangular” scheme U-statistics is studied. Two limit theorems, applicable in different situations, are given. One theorem yields convergence to normal distribution; the other includes Poisson limits laws. Applications statistics based on small interpoint distances sample 1. Introduction. There has been renewed interest theory relating (Hoefi‘ding, 1948) especially focussing degenerate kernels nonnormal limits. See, for instance, Rubin Vitale (1980), Dynkin Mandelbaum (1983), Berman Eagleson (1983). In Section 2 this paper we study establish their normality under suitable conditions. It worth noting that one can obtain laws even with kernels. Other infinitely divisible triangular scheme, including limits, examined next section. distributions discussed 4. These applications, which were source motivation results derived here, suggested by studies us made about spacings. [See, Holst Rao (1981).] They also sparked recent papers Bickel Breiman (1983) Onoyama et al. Theorems yielding setting have proved Weber using martingale approach. His theorems yield same conclusion as Theorem 2.1, but difi‘erent (nonequivalent) few words notation: We write “ —gd ” denote an —gc complete [cf. Loeve (1963), page 178]. Po(}\) mean Np(p., 2) p-dimensional vector p. covariance matrix 2. Asymptotic U-statistics. Let X1, X2,... be sequence independently identically distributed (i.i.d.) random variables. will use X Y two independent Received October 1984; revised April 1985. ‘Formerly J. Rao. AMS 1980 subject classifications. Primary 60F05, 60E07; secondary 62E20. Key phrases. U-statistics, multigraph, laws, distributions, inter-point distances. 1347 ‘T Institute Mathematical Statistics collaborating STOR digitize, preserve, extend access ef‘%}73i ofProbabiIity. 5T0 R ® www.jstor.org