A limit theorem for almost monotone sequences of random variables

In this paper we consider families (Xm,n) of random variables which satisfy a subadditivity condition of the form X0,n+m <= X0,n + Xn,n+m + Yn,n+m, m, n >= 1. The main purpo is to give conditions which are sufficient for the a.e. convergence of ((1/n)X0,n). Restricting ourselves to the case when (X0,n) has certain monotonicity properties, we derive the desired a.e. convergence of ((1/n)X0,n) under moment hypotheses concerning (Ym,n) which are considerably weaker than those in Derriennic [4] and Liggett [15] (in [4,15] no monotonicity assumptions were imposed on (X0,n)). In particular, it turns out that the sequence (E[Y0,n]) may be allowed to grow almost linearly. We also indicate how the obtained convergence results apply to sequences of random sets which have a certain subadditivity property.