Almost subadditive multiparameter ergodic theorems

The purpose of this paper is to extend recent mean as well as a.e. convergence results of Derriennic (1983), Liggett (1985) and Schürger (1986) to multiparameter processes X which satisfy a strong almost subadditivity condition and have certain monotonicity properties. If X is even strongly subadditive, we derive an a.e. limit theorem which extends Tempel'man's (1972) multiparameter pointwise ergodic theorem in the same way as Kingman's (1968) subadditive ergodic theorem extends Birkhoff's ergodic theorem (here, X is not supposed to have any monotonicity properties). We also point out how the results obtained can be used to arrive at analogous results for multiparameter random sets.