A limit theorem for almost monotone sequences of random variables

AbstractIn 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 ((1n)X0,n). Restricting ourselves to the case when (X0,n) has certain monotonicity properties, we derive the desired a.e. convergence of ((1n)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