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Characterization of LIL behavior in Banach space


Authors: Uwe Einmahl and Deli Li
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
Published electronically: July 24, 2008
MathSciNet review: 2434306
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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

DOI: https://doi.org/10.1090/S0002-9947-08-04522-4
Keywords: Law of the iterated logarithm, LIL behavior, Banach space, regularly varying function, sums of i.i.d. random variables, exponential inequalities
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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