Based on two notions of almost subadditivity which were introduced by Derriennic and Schurger, two a.s. limit theorems are proved which both generalize Kingman's subadditive ergodic theorem. These results, being valid under weak moment conditions, are obtained by short proofs. One of these proofs is completely elementary and does not even make use of Birkhoff's ergodic theorem which, instead, is obtained as a by-product. Finally, an improvement of Liggett's a.s. limit theorem is given.
Klaus Schurger. "Almost Subadditive Extensions of Kingman's Ergodic Theorem." Ann. Probab. 19 (4) 1575 - 1586, October, 1991. https://doi.org/10.1214/aop/1176990224