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

 

Forward-convex convergence in probability of sequences of nonnegative random variables


Authors: Constantinos Kardaras and Gordan Žitković
Journal: Proc. Amer. Math. Soc. 141 (2013), 919-929
MSC (2010): Primary 46A16, 46E30, 60A10
Published electronically: July 9, 2012
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Abstract: For a sequence $ (f_n)_{n \in \mathbb{N}}$ of nonnegative random variables, we provide simple necessary and sufficient conditions for convergence in probability of each sequence $ (h_n)_{n \in \mathbb{N}}$ with $ h_n\in \mathrm {conv}(\{f_n,f_{n+1},\dots \})$ for all $ n \in \mathbb{N}$ to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.


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Additional Information

Constantinos Kardaras
Affiliation: Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
Email: kardaras@bu.edu

Gordan Žitković
Affiliation: Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712
Email: gordanz@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11373-5
PII: S 0002-9939(2012)11373-5
Received by editor(s): October 17, 2010
Received by editor(s) in revised form: February 3, 2011, and July 16, 2011
Published electronically: July 9, 2012
Additional Notes: The authors would like to thank Freddy Delbaen and Ted Odell for valuable help, numerous conversations and shared expertise, as well as the anonymous referee for constructive comments and suggestions
Both authors acknowledge partial support by the National Science Foundation, the first author under award number DMS-0908461, and the second author under award number DMS-0706947. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2012 American Mathematical Society