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A new maximal inequality and its applications
Published online by Cambridge University Press: 19 September 2008
Abstract
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 P1 ≤ P2 ≤ … ≤ Pk+1 be positive integers. For a σ-finite measure-preserving system (Ω, β, μ, T) and an a.e. finite β-measurable f denote
Then for any λ > 0 and f ∈ L1(Ω)
We show how the multi-parametric and superadditive versions of the previous equation can be obtained from the corresponding inequality for reversed supermartingales. The possibility of similar theorems for martingales and other sequences is also discussed.
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- Copyright © Cambridge University Press 1992
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