On Decoupling in Banach Spaces

Sonja Cox () and Stefan Geiss ()
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Sonja Cox: University of Amsterdam
Stefan Geiss: University of Jyväskylä

Journal of Theoretical Probability, 2021, vol. 34, issue 3, 1179-1212

Abstract: Abstract 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.

Keywords: Decoupling in Banach spaces; Regular conditional probabilities; Dyadic martingales; Stochastic integration; 60E15; 60H05; 46B09 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-021-01085-6

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