On Recent Developments in the Theory of Set-Indexed Processes
Peter Gaenssler · Contributions to statistics · 1994
Two important processes in probability and statistics are the empirical and partial-sum processes. The purpose of this paper is to présenta unified approach to both types of processes including their multi-variate versions by studying processes Sn=((Sn(C))C ∈ with Sn(C) being defined by $$ {S_n}(C): = \sum\limits_{j\underline < j(n)} {{1_C}({\eta _{nj}}){\xi _{nj}},C \in \ell ,} $$ the ηn j’s being random elements (random locations) in an arbitrary sample space X, the ξ’n js being real-valued random variables (random masses), and where the index set is a Vapnik-Chervonenkis class of subsets of X. Our emphasis is on a asymptotic results (as the sample size n tends to infinity) for the processes Sn such as functional central limit theorems (FCLT) and a uniform law of large numbers (ULLN) (the uniformity being w.r.t.), containing various results for empirical and partial-sum processes as special cases.