Order of approximation in the central limit theorem for associated random variables and a moderate deviation result
Author:
M. Sreehari
Journal:
Theor. Probability and Math. Statist. 98 (2019), 229-242
MSC (2010):
Primary 60E15, 60F10
DOI:
https://doi.org/10.1090/tpms/1073
Published electronically:
August 19, 2019
MathSciNet review:
3824689
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Abstract: An estimate of the order of approximation in the central limit theorem for strictly stationary associated random variables with finite moments of order $q > 2$ is obtained. A moderate deviation result is also obtained. We have a refinement of recent results in Çaǧin et al., J. Korean Statist. Soc. 45 (216), no. 2, 285–294. The order of approximation obtained here is an improvement over the corresponding result of Wood in Ann. Probab. 11 (1983), no. 4, 1042–1047.
References
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- Tonguç Çaǧın, Paulo Eduardo Oliveira, and Nuria Torrado, A moderate deviation for associated random variables, J. Korean Statist. Soc. 45 (2016), no. 2, 285–294. MR 3504944, DOI https://doi.org/10.1016/j.jkss.2015.11.004
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- M. Loève, Probability Theory, I, 4th Ed., Springer, New York, 1977.
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- Thomas E. Wood, A Berry-Esseen theorem for associated random variables, Ann. Probab. 11 (1983), no. 4, 1042–1047. MR 714967
References
- T. Birkel, On the convergence rate in the central limit theorem for associated processes, Ann. Probab. 16 (1988), 1689–1698. MR 958210
- T. Çaǧin, P. E. Oliveira, and N. Torrado, A moderate deviation for associated randon variables, J. Korean Statist. Soc. 45 (2016), 285–294. MR 3504944
- A. Dembo and A. Zeitouni, Large deviation techniques and applications, 2nd ed., Springer, New York, 1998. MR 1619036
- A. N. Frolov, On probabilities of moderate deviations of sums for independent random variables, J. Math. Sci. 127 (2005), 1787–1796. MR 1976757
- F. D. Hollander, Large deviations, Fields Inst. Monographs, Amer. Math. Soc., Providence, Rhodes Island, 2000. MR 1739680
- M. Loève, Probability Theory, I, 4th Ed., Springer, New York, 1977.
- C. M. Newman, Normal fluctuations and FKG-inequality, Commun. Math. Phys. 74 (1980), 119–128. MR 576267
- P. E. Oliveira, Asymptotics for associated random variables, Heidelberg, Springer, 2012. MR 3013874
- A. N. Tikhomirov, On the convergence rate in the central limit theorem for weakly dependent random variables, Theory Probab. Appl. 25 (1980), 790–809. MR 595140
- S. R. S. Varadhan, Large deviations and applications, SIAM, Philadelphia, 1984. MR 758258
- W. Wang, Large deviations for sums of random vectors attracted to operator semi-stable laws, J. Theor. Probab. (2015). MR 3615082
- T. E. Wood, A Berry–Esseen theorem for associated random variables, Ann. Probab. 11 (1983), 1041–1047. MR 714967
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Additional Information
M. Sreehari
Affiliation:
Department of Statistics, The M. S. University of Baroda, Vadodara, 390002, India
Address at time of publication:
6-B, Vrundavan Park, New Sama Road, Vadodara 390024, India
Email:
msreehari03@yahoo.co.uk
Keywords:
Associated random variables,
central limit theorem,
rate of convergence,
Berry–Esséen type bound,
moderate deviations
Received by editor(s):
November 14, 2017
Published electronically:
August 19, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
