On the Longest Edge of the Minimal Spanning Tree

Summary

As part of the effort to (a) robustify the single link clustering method and (b) develop new inference approaches in clustering in the spirit of Bock (1985), we introduce upper and lower bounds for the length M _n of the longest edge of the minimal spanning tree on n iid random variables drawn from a probability measure P, having Lebesgue density f. We prove that with probability converging to 1 as n → ∞:

$$ k_L \cdot \frac{1}{\Delta } \cdot \frac{{\log n}}{n} \leqslant M_n^d \leqslant k_U \cdot \frac{1}{\delta } \cdot \frac{{\log n}}{n} $$

where $\delta = \inf \{ f(x),x\,\, \in \,\,\text{supp(P)\} ,}\,\Delta \,\, = \,\sup \{ f(x),x\,\, \in \,\text{supp(P)}\}$ and k _L , k _U are constants depending only on δsupp(P).