A multiparameter strongly superadditive ergodic theorem
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).
scientific article; zbMATH DE number 3901741
scientific article