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Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Author

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  • Kozek, A.
  • Suchanecki, Z.

Abstract

Let (T, , P) be a probability space, a P-complete sub-[delta]-algebra of and X a Banach space. Let multifunction t --> [Gamma](t), t [set membership, variant] T, have a [circle times operator] (X)-measurable graph and closed convex subsets of X for values. If x(t) [epsilon] [Gamma](t) P-a.e. and y(·) [epsilon] Ep x(·), then y(t) [epsilon] [Gamma](t) P-a.e. Conversely, x(t) [epsilon] F([Gamma](t), y(t)) P-a.e., where F([Gamma](t), y(t)) is the face of point y(t) in [Gamma](t). If X = , then the same holds true if [Gamma](t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.

Suggested Citation

  • Kozek, A. & Suchanecki, Z., 1980. "Multifunctions of faces for conditional expectations of selectors and Jensen's inequality," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 579-598, December.
    • Handle: RePEc:eee:jmvana:v:10:y:1980:i:4:p:579-598
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    Cited by:

    1. Leorato, S., 2009. "A refined Jensen's inequality in Hilbert spaces and empirical approximations," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 1044-1060, May.
    2. Dieter Mussmann, 1988. "Sufficiency and Jensen's inequality for conditional expectations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(4), pages 715-726, December.

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