Functional Law of the Iterated Logarithm and Uniform Central Limit Theorem for Partial-Sum Processes Indexed by Sets

Abstract

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.