A new maximal inequality and its applications
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. © 1992, Cambridge University Press. All rights reserved.
Ergodic Theory and Dynamical Systems
Rosenblatt, J., & Wierdl, M. (1992). A new maximal inequality and its applications. Ergodic Theory and Dynamical Systems, 12 (3), 509-558. https://doi.org/10.1017/S0143385700006921