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August, 1985 The Space $D(A)$ and Weak Convergence for Set-indexed Processes
Richard F. Bass, Ronald Pyke
Ann. Probab. 13(3): 860-884 (August, 1985). DOI: 10.1214/aop/1176992911


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 $r < (\alpha - 1)^{-1}$.


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Richard F. Bass. Ronald Pyke. "The Space $D(A)$ and Weak Convergence for Set-indexed Processes." Ann. Probab. 13 (3) 860 - 884, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0585.60007
MathSciNet: MR799425
Digital Object Identifier: 10.1214/aop/1176992911

Primary: 60B10
Secondary: 60B05 , 60F05 , 60F17

Keywords: $D$-space , central limit theorem , domains of attraction , Empirical processes , partial-sum processes , Stable laws , tightness , weak convergence

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 3 • August, 1985
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