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

Richard F. Bass and Ronald Pyke
The Annals of Probability
Vol. 13, No. 3 (Aug., 1985), pp. 860-884
Stable URL: http://www.jstor.org/stable/2243716
Page Count: 25
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The Space D(A) and Weak Convergence for Set-indexed Processes
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Abstract

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