A strong law for the longest edge of the minimal spanning tree
Let \(f\) be a continuous density on a connected compact support set \(\Omega\) in \(\Re^{d}\) \((d \geq 2)\) that has smooth boundary \(\partial \Omega\). Assume \(f_{0}=\) \(\min_{\Omega}f > 0\) and define \(f_{1}=\) \(\min_{\partial \Omega}f\). Let \(\underline{X}_{n}=\) \((X_1, \dots, X_n)\) be a random sample from \(f\) and define \(Y_n\) to be the smallest \(r\) such that the union of balls of diameter \(r\) centered at \(X_1, \ldots, X_n\) is connected. This \(Y_n\) also represents the maximum edge-length of a minimum spanning tree of the vertex set \(\underline{X}_n\). It is shown that, as \(n \to \infty\), \(n \theta Y_{n}^{d}/\log n \to\) \(\max\{f_{0}^{-1}, 2(1-d^{-1})f_{1}^{-1}\}\) a.s., where \(\theta\) is the volume of the unit ball. This result disproves a conjecture made by \textit{E. Tabakis} [in: From data to knowledge: theoretical and practical aspects of classification, data analysis, and knowledge organization, 222-230 (1996; Zbl 0897.62063)] that the limit is free of \(f\).
scientific article; zbMATH DE number 1359646