Limiting distributions for the maximum of a symmetric function on a random point set
Let \(U_1, U_2,\ldots\) be a sequence of independent random points taking values in a measurable space \((S,\Sigma)\) according to a common probability \(P\) and let \(h:S\times S\to {\mathbb S}\) be a symmetric, Borel\(/\Sigma\times\Sigma\)-measurable function. Let \[ H_n=\max\{h(U_i, U_j): 1\leq i< j\leq n\} \] denote the maximum \(h\)-value over pairs of distinct points from \(U_1, U_2,\ldots,U_n\). Assumptions on the distribution of \(h(U_1,\cdot)\) are provided under which a continuous function of \(H_n\) converges in law to an extreme-value distribution upon suitable rescaling. The work is complementary to that appearing in \textit{M. J. B. Appel, M. J. Klass} and \textit{R. P. Russo} [J. Theor. Probab. 12, No. 1, 27--47 (1999; Zbl 0926.60018)] on the almost-sure limiting behavior of \(H_n\). 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 \({\mathbb R}^d\) 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 pairwise distance among the first \(n\) of i.i.d. uniform points in a compact subset of \({\mathbb R}^d\).
scientific article; zbMATH DE number 5126769
scientific article