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2000/2001 Jensen’s Inequality for Conditional Expectations in Banach Spaces
August M. Zapała
Real Anal. Exchange 26(2): 541-552 (2000/2001).


In this note we present a simple proof of the inequality $\Phi \left( E^{\mathcal{A}}\xi \right) \leq E^{\mathcal{A}}\Phi (\xi )$ a.s. for separable random elements $\xi \mathcal{I}n L_{1}(\Omega ,\mathcal{F},P;X)$ in a Banach space $X,$ where $E^{\mathcal{A}}\left(\cdot\right) $ denotes conditional expectation with respect to the $\sigma $-field $\mathcal{A} \subset \mathcal{F}$, and $\Phi :X\rightarrow \mathbb{R}$ 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 $\left\{ \exp \{c\left\| E^{\mathcal{A}}\xi \right\| ^{2}\}\mathcal{A}_{0}\subseteq \mathcal{A} \subseteq \mathcal{F}\right\} $ is uniformly integrable.


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August M. Zapała. "Jensen’s Inequality for Conditional Expectations in Banach Spaces." Real Anal. Exchange 26 (2) 541 - 552, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 27 June 2008

zbMATH: 1010.60020
MathSciNet: MR1844134

Primary: 26D07 , 28C05 , 28C20 , ‎46G12 , 60B11 , 60E15

Keywords: conditional expectation , convex function , Gaussian random element , Jensen's inequality

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 2 • 2000/2001
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