## The Annals of Probability

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

#### 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

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