Abstract
The classical multidimensional version of Fatou’s lemma (Schmeidler in Proc Am Math Soc 24:300–306, 1970) originally obtained for unconditional expectations and the standard non-negative cone in a finite-dimensional linear space is extended to conditional expectations and general closed pointed cones.
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Notes
A set C in a linear space is called a cone if it contains with any its elements x, y any non-negative linear combination \(\lambda x+\mu y\) (\(\lambda ,\mu \ge 0\)) of these elements. The cone C is called pointed if the inclusions \(x\in C\) and \(-x\in C\) imply \(x=0\).
A measurable space is called standard if it is isomorphic to a Borel subset of a complete separable metric space with the Borel measurable structure.
A set \(C(\omega )\subseteq {\mathbb {R}}^{n}\) is said to depend \({\mathcal {G}}\)-measurably on \(\omega \) if its graph \(\{(\omega ,c):~c\in C(\omega )\}\) belongs to \({\mathcal {G}}\times \mathcal {B(} {\mathbb {R}}^{n})\).
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The authors are grateful to Zvi Artstein for helpful comments.
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Babaei, E., Evstigneev, I.V. & Schenk-Hoppé, K.R. A multidimensional Fatou lemma for conditional expectations. Positivity 25, 1543–1549 (2021). https://doi.org/10.1007/s11117-021-00827-4
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DOI: https://doi.org/10.1007/s11117-021-00827-4
Keywords
- Cones in linear spaces
- Induced partial orderings
- Sequences of random vectors
- Fatou’s lemma
- Conditional expectations