A limit theorem for almost monotone sequences of random variables
We consider families \((X_{m,n})\) of random variables which satisfy a subadditivity condition of the form \(X_{0,n+m}\leq X_{0,n}+X_{n,n+m}+Y_{n,n+m}\), m,n\(\geq 1\). The main purpose of this paper is to give conditions which are sufficient for the a.e. convergence of \(((1/n)X_{0,n}).\) Restricting ourselves to the case when \((X_{0,n})\) has certain monotonicity properties, we derive the desired a.e. convergence of \(((1/n)X_{0,n})\) under moment hypotheses concerning \((Y_{m,n})\) which are considerably weaker than those in \textit{Y. Derriennic}, Ann. Probab. 11, 669-677 (1983; Zbl 0586.28014), and \textit{T. M. Liggett}, ibid. 13, 1279-1285 (1985; Zbl 0579.60023) (in these papers no monotonicity assumptions were imposed on \((X_{0,n}))\). In particular, it turns out that the sequence \((E[Y_{0,n}])\) 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.
scientific article; zbMATH DE number 3984215
scientific article