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Jensen’s inequality for conditional expectations in Banach spaces. (English) Zbl 1010.60020

Summary: We present a simple proof of the inequality \(\Phi(E^{\mathcal A}\xi)\leq E^{\mathcal A}\Phi(\xi)\) a.s. for separable random elements \(\xi\in L_1(\Omega,{\mathcal F},P;X)\) in a Banach space \(X\), where \(E^{\mathcal A}(\cdot)\) denotes conditional expectation with respect to the \(\sigma\)-field \({\mathcal A}\subset{\mathcal F}\), and \(\Phi: X\to\mathbb{R}\) is a convex functional satisfying certain additional assumptions which are less restrictive than known till now. Some consequences of the above result are also discussed; e.g., it is shown that if \(\xi\) is a Gaussian random element in \(X\), then there exists a constant \(0<c<\infty\) such that for each \(\sigma\)-field \({\mathcal A}_0\subset{\mathcal F}\) the family \(\{\exp\{c\|E^{\mathcal A}\xi\|^2\}:{\mathcal A}_0\subseteq{\mathcal A}\subseteq{\mathcal F}\}\) is uniformly integrable.

MSC:

60E15 Inequalities; stochastic orderings
26D07 Inequalities involving other types of functions
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46G12 Measures and integration on abstract linear spaces
60B11 Probability theory on linear topological spaces
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