Let $\{X_j: \mathbf{j} \in J^d\}$ be an array of independent random variables, where $J^d$ denotes the $d$-dimensional positive integer lattice. The main purpose of this paper is to obtain a functional law of the iterated logarithm (LIL) for suitably normalized and smoothed versions of the partial-sum process $S(B) = \sum_{j \in B}X_j$. The method of proof involves the definition of a set-indexed Brownian process, and the embedding of the partial-sum process in this Brownian process. In addition, the LIL is derived for this Brownian process. The method is extended to yield a uniform central limit theorem for the partial-sum process.
Publié le : 1984-02-14
Classification:
Functional law of the iterated logarithm,
partial-sum processes indexed by sets,
central limit theorem,
embedding by stopping times,
invariance principles,
60F15,
60B10,
60J65
@article{1176993371,
author = {Bass, Richard F. and Pyke, Ronald},
title = {Functional Law of the Iterated Logarithm and Uniform Central Limit Theorem for Partial-Sum Processes Indexed by Sets},
journal = {Ann. Probab.},
volume = {12},
number = {4},
year = {1984},
pages = { 13-34},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993371}
}
Bass, Richard F.; Pyke, Ronald. Functional Law of the Iterated Logarithm and Uniform Central Limit Theorem for Partial-Sum Processes Indexed by Sets. Ann. Probab., Tome 12 (1984) no. 4, pp. 13-34. http://gdmltest.u-ga.fr/item/1176993371/