## 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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### References:

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