The almost sure behavior of maximal and minimal multivariate \(k_ n\)- spacings
This paper studies, for large numbers n of independent uniformly distributed random vectors in \([0,1]^ d\), the asymptotic behavior of a very general new notion of maximal and minimal \(k_ n\)-spacings. The notion of a ``maximal \(k_ n\)-spacing'' is here defined relative to certain families \({\mathcal C}\) of subsets of \([0,1]^ d\), the previously studied notion of multivariate \(k_ n\)-spacings having been the one relative to \[ {\mathcal C}\equiv \{[a_ 1,a_ 1+c]\times...\times [a_ d,a_ d+c]: 0\leq a_ j\leq a_ j+c\leq 1\text{ for all } 1\leq j\leq d\}\cup \{\Phi \}. \] For \(k_ n\leq n\), the maximal \(k_ n\)-spacing relative to \({\mathcal C}\) is defined to be \(M_{k_ n,n}\equiv \sup \{\lambda (C):C\in {\mathcal C}\) and \(P_ n(C)<k/n\}\) where \(\lambda\) denotes Lebesgue measure and \(P_ n\) the empirical measure based on the n observed i.i.d. uniform d-vectors. Minimal \(k_ n\)-spacings \(m_{k_ n,n}\) are defined analogously (but with respect to slightly different classes of subsets of \([0,1]^ d).\) The results of the paper are strong laws, and in one instance an iterated-logarithm type law, for the maximal and minimal \(k_ n\)- spacings under \(\epsilon\)-entropy conditions on the classes of sets with respect to which these spacings are defined. The techniques of the proof are based on a study of increments of the empirical measures and their relations to these spacings. It appears that all of the results given here for minimal \(k_ n\)-spacings are completely new, for all d and all classes of sets defining the spacings, whenever \(k_ n>1\).
scientific article; zbMATH DE number 4054705