Abstract
We derive both Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of dimension d of a random field satisfying a weak dependency assumption of projective type: the difference between the expectation of an element of the random field and its conditional expectation given the rest of the field at a distance more than δ is bounded, in distance, by a known decreasing function of δ. The analysis is based on the combination of a multi-scale approximation of random sums by martingale difference sequences, and of a careful decomposition of the domain. The obtained results extend previously known bounds under comparable hypotheses, and do not use the assumption of commuting filtrations.
Acknowledgements
The authors warmly thank Jérome Dedecker for very insightful discussions and comments and Andrey Pilipenko for equally interesting exchanges. The three authors were partially supported by the DFG CRC 1294 – 318763901 ‘Data Assimilation’. The work of A. Carpentier is partially supported by the Deutsche Forschungsgemeinschaft (DFG) Emmy Noether grant MuSyAD (CA 1488/1-1), by the DFG Forschungsgruppe FOR 5381 “Mathematical Statistics in the Information Age – Statistical Efficiency and Computational Tractability”, Project TP 02, by the Agence Nationale de la Recherche (ANR) and the DFG on the French-German PRCI ANR ASCAI CA 1488/4-1 “Aktive und Batch-Segmentierung, Clustering und Seriation: Grundlagen der KI”. GB acknowledges support of the ANR under ANR-19-CHIA-0021-01 “BiSCottE”, and IDEX REC-2019-044. Oleksandr Zadorozhnyi acknowledges Alexander von Humboldt Foundation (Research Group Linkage cooperation Singular diffusions: analytic and stochastic approaches between the University of Potsdam and the Institute of Mathematics of the National Academy of Sciences of Ukraine).
Citation
Gilles Blanchard. Alexandra Carpentier. Oleksandr Zadorozhnyi. "Moment inequalities for sums of weakly dependent random fields." Bernoulli 30 (3) 2501 - 2520, August 2024. https://doi.org/10.3150/23-BEJ1682
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