Limiting Distributions for the Maximum of a Symmetric Function on a Random Point Set

Abstract

Let U1, U2, ... be a sequence of independent random points taking values in a measurable space (S, Σ) according to a common probability P and let \(h:S\times S\rightarrow\)R be a symmetric, Borel/ \(\Sigma\times\Sigma\)-measurable function. Let Hn = max{h(Ui,Uj): 1≤ i < j≤ n} denote the maximum h-value over pairs of distinct points from U1,U2,...,Un. Assumptions on the distribution of h(U1,·) are provided under which a continuous function of Hn converges in law to an extreme-value distribution upon suitable rescaling. The work is complementary to that appearing in Appel et al. (1999) J. Theor. Probab. 12, 27–47. on the almost-sure limiting behavior of Hn. In the first of two examples, the main result applied to the case of i.i.d. points distributed uniformly on the surface of a unit hypersphere in Rd provides the limiting distribution of the maximum pairwise distance (chord length) among the first n of the points. The second example exhibits the limiting distribution of the minimum pair-wise distance among the first n of i.i.d. uniform points in a compact subset of Rd.