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 $r < (\alpha - 1)^{-1}$.
Publié le : 1985-08-14
Classification:
$D$-space,
weak convergence,
central limit theorem,
domains of attraction,
stable laws,
tightness,
partial-sum processes,
empirical processes,
60B10,
60F05,
60F17,
60B05
@article{1176992911,
author = {Bass, Richard F. and Pyke, Ronald},
title = {The Space $D(A)$ and Weak Convergence for Set-indexed Processes},
journal = {Ann. Probab.},
volume = {13},
number = {4},
year = {1985},
pages = { 860-884},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992911}
}
Bass, Richard F.; Pyke, Ronald. The Space $D(A)$ and Weak Convergence for Set-indexed Processes. Ann. Probab., Tome 13 (1985) no. 4, pp. 860-884. http://gdmltest.u-ga.fr/item/1176992911/