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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Characterization of LIL behavior in Banach space
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by Uwe Einmahl and Deli Li PDF
Trans. Amer. Math. Soc. 360 (2008), 6677-6693 Request permission

Abstract:

In a recent paper by the authors a general result characterizing two-sided LIL behavior for real valued random variables has been established. In this paper we look at the corresponding problem in the Banach space setting. We show that there are analogous results in this more general setting. In particular, we provide a necessary and sufficient condition for LIL behavior with respect to sequences of the form $\sqrt {nh(n)}$, where $h$ is from a suitable subclass of the positive, non-decreasing slowly varying functions. To prove these results we have to use a different method. One of our main tools is an improved Fuk-Nagaev type inequality in Banach space which should be of independent interest.
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Additional Information
  • Uwe Einmahl
  • Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
  • Email: ueinmahl@vub.ac.be
  • Deli Li
  • Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1
  • Email: dli@lakeheadu.ca
  • Received by editor(s): October 16, 2006
  • Received by editor(s) in revised form: April 1, 2007
  • Published electronically: July 24, 2008
  • Additional Notes: The first author’s research was supported in part by an FWO Vlaanderen grant.
    The second author’s research was supported in part by an NSERC Canada grant
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 6677-6693
  • MSC (2000): Primary 60B12, 60F15; Secondary 60G50, 60J15
  • DOI: https://doi.org/10.1090/S0002-9947-08-04522-4
  • MathSciNet review: 2434306