The Annals of Probability

The Space $D(A)$ and Weak Convergence for Set-indexed Processes

Richard F. Bass and Ronald Pyke

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In this paper we consider weak convergence of processes indexed by a collection $\mathscr{A}$ of subsets of $I^d$. As a suitable sample space for such processes, we introduce the space $\mathscr{D}(\mathscr{A})$ of set functions that are outer continuous with inner limits. A metric is defined for $\mathscr{D}(\mathscr{A})$ in terms of the graphs of its elements and then we give a sufficient condition for a subset of $\mathscr{D}(\mathscr{A})$ to be compact in this topology. This framework is then used to provide a criterion for probability measures on $\mathscr{D}(\mathscr{A})$ to be tight. As an application, we prove a central limit theorem for partial-sum processes indexed by a family of sets, $\mathscr{A}$, when the underlying random variables are in the domain of normal attraction of a stable law. If $\alpha \in (1, 2)$ denotes the exponent of the limiting stable law, if $r$ denotes the coefficient of metric entropy of $\mathscr{A}$, and if $\mathscr{A}$ satisfies mild regularity conditions, we show that the partial-sum processes converge in law to a stable Levy process provided $r < (\alpha - 1)^{-1}$.

Article information

Ann. Probab., Volume 13, Number 3 (1985), 860-884.

First available in Project Euclid: 19 April 2007

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Primary: 60B10: Convergence of probability measures
Secondary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60B05: Probability measures on topological spaces

$D$-space weak convergence central limit theorem domains of attraction stable laws tightness partial-sum processes empirical processes


Bass, Richard F.; Pyke, Ronald. The Space $D(A)$ and Weak Convergence for Set-indexed Processes. Ann. Probab. 13 (1985), no. 3, 860--884. doi:10.1214/aop/1176992911.

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