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Teoriya Veroyatnostei i ee Primeneniya, 1976, Volume 21, Issue 2, Pages 334–347
(Mi tvp3350)
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This article is cited in 47 scientific papers (total in 47 papers)
Regular conditional expectations of correspondences
E. B. Dynkin, I. V. Evstigneev Moscow
Abstract:
Let $f(\omega,z)$ be random variables depending on a parameter $z$. We construct a function $g(\omega,z)$ with the following property: $g(\omega,\varphi(\omega))$ is the conditional expectation with respect to $\mathscr G$, of $f(\omega,\varphi(\omega))$ for each $\mathscr G$-measurable $\varphi$. All the functions $g$ with the above property are indistinguishable (i. e. coincide for all $z$ almost surely). We call them regular conditional expectations of $f$.
We also study functions with values in the set of all closed non-empty subsets of a finite-dimensional vector space. Basic theorems about regular conditional expectations are extended for all such set-valued functions (we call them correspondences).
Until recently conditional expectations were considered only for convex-valued correspondences (see e. g. [3]). To eliminate this restriction, we use the notion of $\mathscr G$-atoms introduced by V. A. Rokhlin [13] (see also [11]), consider a decomposition of the space into disjoint $\mathscr G$-atoms and a non-atomic part and prove for the latter a generalization of well-known theorem of Lyapunov.
Received: 07.10.1974
Citation:
E. B. Dynkin, I. V. Evstigneev, “Regular conditional expectations of correspondences”, Teor. Veroyatnost. i Primenen., 21:2 (1976), 334–347; Theory Probab. Appl., 21:2 (1977), 325–338
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