Abstract
A maximal inequality is an inequality which involves the (absolute) supremum or the running maximum of a stochastic process . We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type – including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process –, strong Markov processes and Gaussian processes. Using the Burkholder–Davis–Gundy inequalities we also discuss some relations between maximal estimates in probability and the Hardy–Littlewood maximal functions from analysis.
Funding Statement
Financial support through the DFG-NCN Beethoven Classic 3 project SCHI419/11-1 & NCN 2018/31/G/ST1/02252 is gratefully acknowledged.
Acknowledgments
We thank the editor-in-chief of Probability Surveys for the invitation to contribute to this journal and for the careful handling of the submission. The efforts and advice of two anonymous reviewers are much appreciated. We are deeply indebted to our colleagues and students, Dr. David Berger, Dr. Wojciech Cygan, Mr. Mustafa Hamadi and Ms. Cailing Li, for proofreading and helpful comments. A substantial part of this paper was written while the first-named author was research associate at the Mathematics Department of TU Dresden. She would like to thank the department for the good working conditions.
Citation
Franziska Kühn. René L. Schilling. "Maximal inequalities and some applications." Probab. Surveys 20 382 - 485, 2023. https://doi.org/10.1214/23-PS17
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