## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 1 (1984), 13-34.

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

Richard F. Bass and Ronald Pyke

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 1 (1984), 13-34.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993371

**Digital Object Identifier**

doi:10.1214/aop/1176993371

**Mathematical Reviews number (MathSciNet)**

MR723727

**Zentralblatt MATH identifier**

0572.60037

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60B10: Convergence of probability measures 60J65: Brownian motion [See also 58J65]

**Keywords**

Functional law of the iterated logarithm partial-sum processes indexed by sets central limit theorem embedding by stopping times invariance principles

#### Citation

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. 12 (1984), no. 1, 13--34. doi:10.1214/aop/1176993371. https://projecteuclid.org/euclid.aop/1176993371