Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Upper bounds for series involving moderate and small deviations

Author: Aurel Spataru
Journal: Proc. Amer. Math. Soc. 138 (2010), 2601-2606
MSC (2010): Primary 60G50; Secondary 60E15, 60F15
Published electronically: February 26, 2010
MathSciNet review: 2607890
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Abstract: Let $ X,~X_{1},~X_{2},...$ be i.i.d. random variables with $ 0<EX^{2}=\sigma ^{2}<\infty $ and $ EX=0,$ and set $ S_{n}=X_{1}+\cdots+X_{n}.$ We prove Paley-type inequalities for series involving probabilities of moderate deviations $ P(\left\vert S_{n}\right\vert \geq \lambda \sqrt{n\log n}),$ $ \lambda >0,$ and probabilities of small deviations $ P(\left\vert S_{n}\right\vert \geq$ $ \lambda \sqrt{n\log \log n})$, $ \lambda >\sigma \sqrt{2}.$

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Additional Information

Aurel Spataru
Affiliation: Casa Academiei Romane, Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie, nr. 13, 76100 Bucharest, Romania

Keywords: Tail probabilities of sums of i.i.d. random variables, Paley-type inequalities, moderate deviations, small deviations, law of the iterated logarithm.
Received by editor(s): June 7, 2009
Received by editor(s) in revised form: October 9, 2009
Published electronically: February 26, 2010
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.