ON THE h-DIAMETER OF A RANDOM POINT SET

Let S ∑ be a measurable space and let h : S S R be a real symmetric Borel/∑ ∑measurable function on S S. Let B be a non-empty measurable subset in S and let μ be a probability measure supported on the restriction of the measurable space S ∑ to B. Let B have finite h-diameter h ess sup h u v : u v B ∞ Let U U1 U2 .... be a sequence of independent random points taking values in B according to μ and let Hn max h Ui Uj : 1 i j n denote the h-diameter of the set Ui i 1 n , the maximum pair-wise h-distance among the first n points. A theoretical framework is provided from which one may deduce the weak convergence of H n, upon suitable centering and rescaling, to an extreme-value distribution. The sufficient condition provided herein is quite different from that of Appel, et al. [1]. Several applications of the theory are provided.