## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 4 (1987), 1569-1585.

### A Two-Parameter Maximal Ergodic Theorem with Dependence

#### Abstract

Let $X_1, X_2, \ldots$ and $Y_1, Y_2, \ldots$ be independent sequences of i.i.d. $U(0, 1)$ random variables. We characterize completely those Borel functions $F$ on $\lbrack 0, 1\rbrack^2$ for which the strong law of large numbers and the maximal ergodic theorem hold for the doubly indexed family $(1/nm)\sum_{i \leq n, j \leq m}F(X_i, Y_j)$.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 4 (1987), 1569-1585.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991994

**Digital Object Identifier**

doi:10.1214/aop/1176991994

**Mathematical Reviews number (MathSciNet)**

MR905349

**Zentralblatt MATH identifier**

0656.28011

**JSTOR**

links.jstor.org

**Subjects**

Primary: 28D05: Measure-preserving transformations

Secondary: 60G60: Random fields

**Keywords**

Maximal ergodic theorem strong law of large numbers two-parameter martingales decoupling

#### Citation

McConnell, Terry R. A Two-Parameter Maximal Ergodic Theorem with Dependence. Ann. Probab. 15 (1987), no. 4, 1569--1585. doi:10.1214/aop/1176991994. https://projecteuclid.org/euclid.aop/1176991994