Bass, Richard F.; Pyke, Ronald The space D(A) and weak convergence for set-indexed processes. (English) Zbl 0585.60007 Ann. Probab. 13, 860-884 (1985). A topology is introduced for the space D(A) of functions which are outer continuous with inner limits indexed by a family of Borel subsets of the d-dimensional unit cube. The space is a natural extension of D[0,1] as a range space of sample paths for processes indexed by a family of sets. Weak convergence of such processes is considered. The topology is similar in spirit to the \(M_ 2\) topology of Skorokhod for D[0,1] in that the distance between two functions is defined by the Hausdorff distance between their graphs. A central limit theorem for partial-sum processes indexed by a family of sets is established. Reviewer: D.P.Kennedy Cited in 1 ReviewCited in 14 Documents MSC: 60B10 Convergence of probability measures 60F17 Functional limit theorems; invariance principles 60B05 Probability measures on topological spaces Keywords:Weak convergence; topology of Skorokhod; Hausdorff distance; central limit theorem PDFBibTeX XMLCite \textit{R. F. Bass} and \textit{R. Pyke}, Ann. Probab. 13, 860--884 (1985; Zbl 0585.60007) Full Text: DOI