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

 

Random series in $ L^p(X,\Sigma,\mu)$ using unconditional basic sequences and $ l^p$ stable sequences: A result on almost sure almost everywhere convergence


Authors: Juan M. Medina and B. Cernuschi-Frías
Journal: Proc. Amer. Math. Soc. 135 (2007), 3561-3569
MSC (2000): Primary 46B20, 60B11, 46B09; Secondary 40A05
Published electronically: June 21, 2007
MathSciNet review: 2336571
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Abstract: Here we study the almost sure almost everywhere convergence of random series of the form $ \sum_{i=1}^{\infty} {a_i f_i}$ in the Lebesgue spaces $ L^p(X,\Sigma,\mu)$, where the $ a_i$'s are centered random variables, and the $ f_i$'s constitute an unconditional basic sequence or an $ l^p$ stable sequence. We show that if one of these series converges in the norm topology almost surely, then it converges almost everywhere almost surely.


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Additional Information

Juan M. Medina
Affiliation: Facultad de Ingeniería, Paseo Colón 850 (1063), Capital Federal, Depto. de Mathemática, Universidad de Buenos Aires and Instituto Argentino de Matemática, Conicet, Argentina
Email: jmedina@fi.uba.ar

B. Cernuschi-Frías
Affiliation: Facultad de Ingeniería, Universidad de Buenos Aires and Instituto Argentino de Matemática, Conicet, Argentina
Email: bcf@ieee.org

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08870-3
PII: S 0002-9939(07)08870-3
Keywords: Unconditional basic sequence, almost sure convergence, random series.
Received by editor(s): November 15, 2005
Received by editor(s) in revised form: August 7, 2006
Published electronically: June 21, 2007
Additional Notes: This work was partially supported by the Universidad de Buenos Aires, grant No. I028, and the Consejo Nacional de Investigaciones Científicas y Técnicas, Conicet, Argentina
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.