The Annals of Probability

Convexity and Conditional Expectations

J. Pfanzagl

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Abstract

If a $n$-dimensional function is with probability one in a convex set, the same holds true for the conditional expectation (with respect to any sub-$\sigma$-field). An extreme point of this convex set can be assumed by the conditional expectation only if it is assumed by the original function and if this function is partially measurable with respect to the conditioning sub-$\sigma$-field. These results are used to prove Jensen's inequality for conditional expectations of $n$-dimensional functions, and to give a condition for strict inequality.

Article information

Source
Ann. Probab., Volume 2, Number 3 (1974), 490-494.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996665

Digital Object Identifier
doi:10.1214/aop/1176996665

Mathematical Reviews number (MathSciNet)
MR358893

Zentralblatt MATH identifier
0285.60002

JSTOR
links.jstor.org

Subjects
Primary: 62B99: None of the above, but in this section
Secondary: 52A40: Inequalities and extremum problems

Keywords
Conditional expectations convex sets

Citation

Pfanzagl, J. Convexity and Conditional Expectations. Ann. Probab. 2 (1974), no. 3, 490--494. doi:10.1214/aop/1176996665. https://projecteuclid.org/euclid.aop/1176996665


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