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Limit theorems for a triangular scheme of U-statistics with applications to inter-point distances. (English) Zbl 0604.60023
Let \(X_ 1,X_ 2,..\). be a sequence of independent and identically distributed random variables and for each \(n=2,3,..\). let \(f_ n(x,y)\) be a measurable, symmetric function of two variables. This paper provides conditions under which the statistic \[ U_ n=\sum_{1\leq i<j\leq n}f_ n(X_ i,X_ j), \] suitably normed, converges in distribution to an infinitely divisible law. Convergence to a normal distribution is considered in some detail; the authors present results similar to those of N. C. Weber [Math. Proc. Camb. Philos. Soc. 94, 307-313 (1983; Zbl 0563.60025)] but under different conditions. Applications to interpoint distance statistics are discussed.
Reviewer: N.Weber

MSC:
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
62E20 Asymptotic distribution theory in statistics
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