Uniform pointwise ergodic theorems for classes of averaging sets and multiparameter subadditive processes

Recently, Bass and Pyke proved a strong law of large numbers for d-dimensional arrays of i.i.d. random variables in which the a.e.-convergence was uniform over a large family of averaging sets. Using different arguments from ergodic theory, we extend this result to multiparameter subadditive processes. Even without the uniformity statement this yields convergence a.e. for more general averaging sequences than those considered by Akcoglu and Krengel.