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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic behavior of $ p$-predictions for vector valued random variables


Authors: Juan A. Cuesta and Carlos Matrán
Journal: Proc. Amer. Math. Soc. 100 (1987), 716-720
MSC: Primary 60G25; Secondary 41A65, 60B99
MathSciNet review: 894443
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Abstract: Let $ (\Omega ,\sigma ,\mu )$ be a probability space and let $ X$ be a $ B$-valued $ \mu $-essentially bounded random variable, where $ (B,\left\Vert {} \right\Vert)$ is a uniformly convex Banach space. Given $ \alpha $, a sub-$ \sigma $-algebra of $ \sigma $, the $ p$-prediction $ 1 < p < \infty $ of $ X$ is defined as the best $ {L_p}$-approximation to $ X$ by $ \alpha $-measurable random variables.

The paper proves that the Pólya algorithm is successful, i.e. the $ p$-prediction converges to an "$ \infty $-prediction" as $ p \to \infty $. First the proof is given for $ p$-means ($ p$-predictions given the trivial $ \sigma $-algebra), and the general case follows from the characterization of the $ p$-prediction in terms of the $ p$-mean of the identity in $ B$ with respect to a regular conditional probability. Notice that the problem was treated in [7], but the proof is not satisfactory (as pointed out in [4]).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0894443-3
PII: S 0002-9939(1987)0894443-3
Article copyright: © Copyright 1987 American Mathematical Society