A limit theorem for almost monotone sequences of random variables

Klaus Schürger

Stochastic Processes and their Applications, 1986, vol. 21, issue 2, 327-338

Abstract: In this paper we consider families (Xm,n) of random variables which satisfy a subadditivity condition of the form X0,n+m = 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.

Keywords: L1-convergence; a.e.; convergence; subadditive; ergodic; theorem; almost; subadditive; sequence; superstationary; sequence; percolation; entropy; random; sets (search for similar items in EconPapers)
Date: 1986
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