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A new maximal inequality and its applications. (English) Zbl 0757.28015
There is a maximal inequality on the integers which implies not only the classical ergodic maximal inequality and certain maximal inequalities for moving averages and differentiation theory, but it also has the following consequence:
Let \(P_ 1\leq P_ 2\leq\dots\leq P_{k+1}\) be positive integers. For a \(\sigma\)-finite measure-preserving system \((\Omega,\beta,\mu,T)\) and an a.e finite \(\beta\)-measurable \(f\) denote \(M_ mf(\omega)={1\over m}\sum^{m-1}_{u=0}f(T^ u\omega)\); \(m=1,2,\dots\;\). Then for any \(\lambda>0\) and \(f\in L^ 1(\Omega)\) \[ \sum^ k_{n=1}\mu\left(\sup_{P_ n\leq m\leq P_{n+1}}| M_ mf(\omega)|>\lambda n\right)\leq{2\over \lambda}\| f\|_{L^ 1(\Omega)}. \tag{1} \] We show how the multi-parametric and superadditive versions of (1) can be obtained from the corresponding inequality for reversed super-martingales. The possibility of similar theorems for martingales and other sequences is also discussed.
Reviewer: J.M.Rosenblatt

28D05 Measure-preserving transformations
37A99 Ergodic theory
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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Full Text: DOI
[1] DOI: 10.2307/2039829 · Zbl 0278.28011
[2] DOI: 10.2307/1970516 · Zbl 0186.20503
[3] Krengel, Ergodic Theorems 6 (1985)
[4] DOI: 10.1016/0001-8708(84)90038-0 · Zbl 0546.42017
[5] DOI: 10.2307/1994170 · Zbl 0142.14802
[6] DOI: 10.1214/aoms/1177704268 · Zbl 0209.49503
[7] DOI: 10.1007/BF01673506 · Zbl 0398.28021
[8] Bellow, Ergod. Th. & Dynam. Sys. 10 pp 43– (1990)
[9] DOI: 10.1073/pnas.59.2.349 · Zbl 0185.21806
[10] Kakutani, Proc. Imp. Akad. 15 pp 165– (1939) · Zbl 0021.41201
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