Upper bounds for series involving moderate and small deviations
Author:
Aurel Spataru
Journal:
Proc. Amer. Math. Soc. 138 (2010), 26012606
MSC (2010):
Primary 60G50; Secondary 60E15, 60F15
Published electronically:
February 26, 2010
MathSciNet review:
2607890
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Additional Information
Abstract: Let be i.i.d. random variables with and and set We prove Paleytype inequalities for series involving probabilities of moderate deviations and probabilities of small deviations ,
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 Chow, Y. S. and Lai, T. L. (1978). Paleytype inequalities and convergence rates related to the law of large numbers and extended renewal theory. Z. Wahrscheinlichkeitstheorie verw. Gebiete 45, 119. MR 507969 (80c:60048)
 2.
 Davis, J. A. (1968). Convergence rates for the law of the iterated logarithm. Ann. Math. Statist. 39, 14791485. MR 0253411 (40:6626)
 3.
 Davis, J. A. (1968). Convergence rates for probabilities of moderate deviations. Ann. Math. Statist. 39, 20162028. MR 0235599 (38:3903)
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 Erdős, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286291. MR 0030714 (11:40f)
 5.
 Erdős, P. (1950). Remark on my paper ``On a theorem of Hsu and Robbins''. Ann. Math. Statist. 21, 138. MR 0032970 (11:375b)
 6.
 Gut, A. and Spătaru, A. (2000). Precise asymptotics in the BaumKatz and Davis laws of large numbers. J. Math. Anal. Appl. 248, 233246. MR 1772594 (2001g:60064)
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 Gut, A. and Spătaru, A. (2000). Precise asymptotics in the law of the iterated logarithm. Ann. Probab. 28, 18701883. MR 1813846 (2001m:60100)
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 Hsu, P. L. and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33, 2531. MR 0019852 (8:470e)
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 Katz, M. L. (1963). The probability in the tail of a distribution. Ann. Math. Statist. 34, 312318. MR 0144369 (26:1914)
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 Loève, M. (1977). Probability Theory. I, 4th edition. SpringerVerlag, Berlin. MR 0651017 (58:31324a)
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 Nagaev, S. V. (1965). Some limit theorems for large deviations. Theor. Probab. Appl. 10, 214235. MR 0185644 (32:3106)
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 Petrov, V. V. (1975). Sums of Independent Random Variables. SpringerVerlag, Berlin. MR 0388499 (52:9335)
 13.
 Pruss, A. R. (1997). A twosided estimate in the HsuRobbinsErdős law of large numbers. Stochastic Process. Appl. 70, 173180. MR 1475661 (99c:60100)
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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
DOI:
http://dx.doi.org/10.1090/S0002993910102470
PII:
S 00029939(10)102470
Keywords:
Tail probabilities of sums of i.i.d. random variables,
Paleytype 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.
