A Precise Asymptotic Behaviour of the Large Deviation Probabilities for Weighted Sums

Gooty Divanji, Kokkada Vidyalaxmi
DOI: 10.4236/am.2011.29163                

Abstract

Let {X_n, n ≥ 1} be a sequence of independent and identically distributed positive valued random variables with a common distribution function F. When F belongs to the domain of partial attraction of a semi stable law with index α, 0 < α < 1, an asymptotic behavior of the large deviation probabilities with respect to properly normalized weighted sums have been studied and in support of this we obtained Chover’s form of law of iterated logarithm.

Keywords

Large Deviations, Law of Iterated Logarithm, Semi-Stable Law, Domain of Partial Attraction, Weighted Sums

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	V. M. Kruglov, “On the Extension of the Class of Stable Distributions,” Theory of Probability and Its Applications, Vol. 17, 1972, pp. 685-694. doi:10.1137/1117081
[2]	C. C. Heyde, “A Contribution to the Theory of Large Deviations for Sums of Independent Random Variables,” Probability Theory and Related Fields, Vol. 7, No. 5, 1967, pp. 303-308. doi:10.1007/BF00535016
[3]	C. C. Heyde, “On Large Deviation Problems for Sums of Random Variables Which Are Not Attracted to the Normal Law,” The Annals of Mathematical Statistics, Vol. 38, No. 5, 1967, pp. 1575-1578. doi:10.1214/aoms/1177698712
[4]	C. C. Heyde, “On Large Deviation Probabilities in the Case of Attraction to a Non-Normal Stable Law,” Sankhyā: The Indian Journal of Statistics, Series A, Vol. 30, 1968, No. 3, pp. 253-258.
[5]	J. Chover, “A Law of the Iterated Logarithm for Stable Summands,” Proceedings of the American Mathematical, Vol. 17, No. 2, 1966, pp. 441-443.
[6]	L. Peng and Y. C. Qi, “Chover-Type Laws of the Iterated Logarithm for Weighted Sums,” Statistics and Probability Letters, Vol. 65, No. 4, 2003, pp. 401-410. doi:10.1016/j.spl.2003.08.009
[7]	G. Divanji, “Law of the Iterated Logarithm for Subsequences of Partial Sums Which are in the Domain of Partial Attraction of Semi Stable Law,” Probability and Mathematical Statistics, Vol. 24, pp. 433-442.
[8]	G. Divanji and R. Vasudeva. “Tail Behavior of Distributions in the Domain of Partial Attraction and Some Related Iterated Logarithm Laws,” Sankhyā: The Indian Journal of Statistics, Series A, Vol. 51, No. 2, pp. 196-204.
[9]	D. Drasin and E. Seneta, “A Generalization of Slowly Varying Functions,” Proceedings of the American Mathe- matical Society, Vol. 96, No. 3, pp. 470-471.
[10]	R. Vasudeva and G. Divanji, “Law of Iterated Logarithm for Random Subsequences,” Statistics & Probability Letters, Vol. 12, No. 3, 1991, pp. 189-194. doi:10.1016/0167-7152(91)90076-4
[11]	R. Vasudeva, “A Log Log Law for Abel’s Sum,” Acta Mathematica Academiae Scientiarum Hungaricae Tomus Vol. 32, No. 3-4, 1978, pp. 205-215.
[12]	W. Feller, “An Introduction to Probability Theory and Its Application,” Wiley & Sons, New York. 1966.