A limit theorem for almost monotone sequences of random variables
type: Item
instance of: scholarly article
title: A limit theorem for almost monotone sequences of random variables
zbMATH Open document ID: 0608.60027
DOI: 10.1016/0304-4149(86)90104-3
published in: Stochastic Processes and their Applications
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 3984215
zbMATH Keywords: percolation
zbMATH Keywords: monotonicity properties
zbMATH Keywords: random sets
zbMATH Keywords: subadditivity property
author: Klaus Schürger
MaRDI profile type: Publication
full work available at URL: 0304-4149(86)90104-3
OpenAlex ID: W2062075389
cites work: Q4167341
cites work: Law of Large Numbers for Random Sets and Allocation Processes
cites work: A strong law of large numbers for random compact sets
cites work: Un théorème ergodique presque sous-additif. (An almost subadditive ergodic theorem)
cites work: On the growth of one dimensional contact processes
cites work: Oriented percolation in two dimensions
cites work: Q5732774
cites work: Postulates for subadditive processes
cites work: Q5573847
cites work: Subadditive ergodic theory
cites work: Stochastic partial ordering
cites work: Stochastic inequalities on partially ordered spaces
cites work: A simple proof of some ergodic theorems
cites work: Q4112681
cites work: An improved subadditive ergodic theorem
cites work: Q5185817
cites work: Q4084463
cites work: Q3037731
cites work: Ergodic theorems for subadditive superstationary families of convex compact random sets
cites work: On Derriennic's almost subadditive ergodic theorem
cites work: First-passage percolation on the square lattice
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description: scientific article; zbMATH DE number 3984215