## The Annals of Probability

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

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

Richard F. Bass and Ronald Pyke

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992911

**Digital Object Identifier**

doi:10.1214/aop/1176992911

**Mathematical Reviews number (MathSciNet)**

MR799425

**Zentralblatt MATH identifier**

0585.60007

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aop/1176992911