On decoupling in Banach spaces

2018/05/31 by Cox, Sonja, Geiss, Stefan
#46B09 #60E15 #60H05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1805.12377

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 same conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not effect the decoupling properties (e.g., the constants involved). As special case we deal with one-sided moment inequalities when decoupling dyadic (i.e., Paley-Walsh) martingales. We establish the decoupling constant of ℝ^d with the l^∞-norm. As an example of an application, we demonstrate 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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