Asymptotic of power-weighted Euclidean functionals
For i.i.d. random points \(X_1,\ldots ,X_n\) in \({\mathbb R}^d\) and \(0<p<1\), let \(L(X_1,\ldots ,X_n;p)\) be the minimum of \(\sum _{j=1}^n |X_{\pi (j+1)}-X_{\pi (j)} |^p\) over all permutations \(\pi \) of \(\{ 1,\ldots ,n\}\) (minimal path of the `travelling salesman'). The author shows that, under some regularity conditions posed on the distribution of the \(X_i\)'s, \(n^{(d-p)/d}L(X_1,\ldots ,X_n;p)\) converges to a given constant a.s. and in \(L^1\) (an analogous result was known earlier for \(p\geq 1\)). In fact, a more general result is proved, which yields as particular cases also the asymptotic behavior of the length of an optimal matching and of the length of a minimal spanning tree on \(X_1,\ldots ,X_n\).
scientific article; zbMATH DE number 1554276