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 be i.i.d. random variables with and and set We prove Paley-type inequalities for series involving probabilities of moderate deviations and probabilities of small deviations ,

**1.**Y. S. Chow and T. L. Lai,*Paley-type inequalities and convergence rates related to the law of large numbers and extended renewal theory*, Z. Wahrsch. Verw. Gebiete**45**(1978), no. 1, 1–19. MR**507969**, 10.1007/BF00635960**2.**James Avery Davis,*Convergence rates for the law of the iterated logarithm*, Ann. Math. Statist.**39**(1968), 1479–1485. MR**0253411****3.**James Avery Davis,*Convergence rates for probabilities of moderate deviations*, Ann. Math. Statist.**39**(1968), 2016–2028. MR**0235599****4.**P. Erdös,*On a theorem of Hsu and Robbins*, Ann. Math. Statistics**20**(1949), 286–291. MR**0030714****5.**P. Erdös,*Remark on my paper “On a theorem of Hsu and Robbins.”*, Ann. Math. Statistics**21**(1950), 138. MR**0032970****6.**Allan Gut and Aurel Spătaru,*Precise asymptotics in the Baum-Katz and Davis laws of large numbers*, J. Math. Anal. Appl.**248**(2000), no. 1, 233–246. MR**1772594**, 10.1006/jmaa.2000.6892**7.**Allan Gut and Aurel Spătaru,*Precise asymptotics in the law of the iterated logarithm*, Ann. Probab.**28**(2000), no. 4, 1870–1883. MR**1813846**, 10.1214/aop/1019160511**8.**P. L. Hsu and Herbert Robbins,*Complete convergence and the law of large numbers*, Proc. Nat. Acad. Sci. U. S. A.**33**(1947), 25–31. MR**0019852****9.**Melvin L. Katz,*The probability in the tail of a distribution*, Ann. Math. Statist.**34**(1963), 312–318. MR**0144369****10.**Michel Loève,*Probability theory. I*, 4th ed., Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, Vol. 45. MR**0651017****11.**S. V. Nagaev,*Some limit theorems for large deviations*, Teor. Verojatnost. i Primenen**10**(1965), 231–254 (Russian, with English summary). MR**0185644****12.**V. V. Petrov,*Sums of independent random variables*, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. MR**0388499****13.**Alexander R. Pruss,*A two-sided estimate in the Hsu-Robbins-Erdős law of large numbers*, Stochastic Process. Appl.**70**(1997), no. 2, 173–180. MR**1475661**, 10.1016/S0304-4149(97)00068-9

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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:
https://doi.org/10.1090/S0002-9939-10-10247-0

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.