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 $(1/nm)\sum_{i \leq n, j \leq m}F(X_i, Y_j)$.
Citation
Terry R. McConnell. "A Two-Parameter Maximal Ergodic Theorem with Dependence." Ann. Probab. 15 (4) 1569 - 1585, October, 1987. https://doi.org/10.1214/aop/1176991994
Information