The space D(A) and weak convergence for set-indexed processes. (English) Zbl 0585.60007

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


60B10 Convergence of probability measures
60F17 Functional limit theorems; invariance principles
60B05 Probability measures on topological spaces
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