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Extremes on trees. (English) Zbl 1096.60009

Summary: This paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on \({\mathbf X}_1,\dots,{\mathbf X}_{N_n}\) where \({\mathbf X}_1,{\mathbf X}_2,\dots\) are i.i.d. in \(\operatorname{Re}^2\) with distribution \(F\) and \(N_n\) is independent of the \({\mathbf X}_i\) and satisfies \(N_n/n\to_p 1\). A new approach based on spatial blocking and a locally orthogonal coordinate system is developed to treat cases for which \(F\) has unbounded support. The general results are applied to a number of special cases, including elliptically contoured distributions, distributions with independent Weibull-like margins and distributions with parallel level curves.

MSC:

60D05 Geometric probability and stochastic geometry
60F05 Central limit and other weak theorems
60G70 Extreme value theory; extremal stochastic processes
05C05 Trees
05C80 Random graphs (graph-theoretic aspects)
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