On Decoupling in Banach Spaces
Cox, S., & Geiss, S. (2021). On Decoupling in Banach Spaces. Journal of Theoretical Probability, 34(3), 1179-1212. https://doi.org/10.1007/s10959-021-01085-6
Published inJournal of Theoretical Probability
© The Author(s) 2021
We consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley–Walsh) martingales and show that Burkholder–Davis–Gundy-type inequalities for stochastic integrals of X-valued processes can be obtained from decoupling inequalities for X-valued dyadic martingales.
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Related funder(s)Academy of Finland
Funding program(s)Academy Project, AoF
Additional information about fundingThe first author is supported by the research program VENI Vernieuwingsimpuls with Project Number 639.031.549, which is financed by the Netherlands Organization for Scientific Research (NWO). The second author is supported by the project Stochastic Analysis and Nonlinear Partial Differential Equations, Interactions and Applications of the Academy of Finland with Project Number 298641. The authors wish to thank Mark Veraar, Peter Spreij, and an anonymous referee. The first author would also like to thank Lotte Meijer. Open access funding provided by University of Jyväskylä (JYU). ...
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