## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 1 (1984), 268-271.

### A Strong Law of Large Numbers for Partial-Sum Processes Indexed by Sets

Richard F. Bass and Ronald Pyke

#### Abstract

Let $J = \{1, 2, \cdots\}^d$ and let $\{X_j, \mathbf{j} \in J\}$ be iid with finite mean. Let $S(nA)$ be the sum of those $X_j$'s for which $\mathbf{j}/n \in A$. It is proved in this paper that $S(\cdot)$ satisfies a strong law of large numbers that is uniform over $A \in \mathscr{A}$, where $\mathscr{A}$ is a family of subsets of $\lbrack 0, 1\rbrack^d$ satisfying a mild condition.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 1 (1984), 268-271.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993390

**Digital Object Identifier**

doi:10.1214/aop/1176993390

**Mathematical Reviews number (MathSciNet)**

MR723746

**Zentralblatt MATH identifier**

0543.60036

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G99: None of the above, but in this section

**Keywords**

Strong law of large numbers partial-sum processes processes indexed by sets

#### Citation

Bass, Richard F.; Pyke, Ronald. A Strong Law of Large Numbers for Partial-Sum Processes Indexed by Sets. Ann. Probab. 12 (1984), no. 1, 268--271. doi:10.1214/aop/1176993390. https://projecteuclid.org/euclid.aop/1176993390