A multiparameter strongly superadditive ergodic theorem
type: Item
instance of: scholarly article
title: A multiparameter strongly superadditive ergodic theorem
zbMATH Open document ID: 0565.60033
DOI: 10.1007/BF01246940
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173563
review text: Let \((X,{\mathcal F},\mu)\) be a \(\sigma\)-finite measure space and \(T_ 1,T_ 2\) two commuting positive contractions on both \(L_ 1\) and \(L_{\infty}\). According to the definition of \textit{R. T. Smythe} [Ann. Probab. 4, 772-782 (1976; Zbl 0339.60022)], the family \(\{F(m,n):\quad m,n\in {\mathbb{N}}\}\) of \(L_ p\) random variables is called strongly superadditive (with respect to the operators \(T_ i\), \(i=1,2)\) if for all n,m,r, and s: \[ F(n+m,\quad r+s)\geq F(n,r)+T^ n_ 1F(m,r+s)+T^ r_ 2F(n+m,s)-T^ n_ 1T^ r_ 2F(m,s), \] \[ F(n,0)=F(0,n)=0,\quad and\quad \sup_{n,m}(nm)^{-1}\| F(n,m)\| p<\infty. \] The theorems of the present paper concern additional conditions on \(T_ i\) and \(\{F(m,n)\}\) under which \(n^{-2}F(n,n)\) converges a.e. or in \(L_ p\) as \(n\to \infty\), extending work of \textit{A. Brunel}, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 9(1973), 327-343 (1974; Zbl 0272.47007) and \textit{M. A. Akcoglu} and L. Sucheston, see Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, 269-278 (1978; Zbl 0386.60045).
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 28D15
zbMATH DE Number: 3901741
zbMATH Keywords: commuting positive contractions
zbMATH Keywords: strongly superadditive
author: Richard Emilion
author: Bachar Hachem
reviewed by: Eric V. Slud
MaRDI profile type: Publication
cites work: A Pointwise Ergodic Theorem in Lp-Spaces
cites work: Ergodic theorems for superadditive processes.
cites work: On convergence of iterates of positive contractions in L\(_p\) spaces
cites work: A ratio ergodic theorem for superadditive processes
cites work: Subadditive mean ergodic theorems
cites work: Q3911152
cites work: Subadditive ergodic theory
cites work: Multiparameter subadditive processes
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description: scientific article; zbMATH DE number 3901741