Abstract
In this paper, we obtain results on precise large deviations for non–random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, three precise large–deviation probabilities with different centering numbers are equivalent to each other. Furthermore, we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve the corresponding results of Ng, Tang, Yan and Yang (J. Appl. Prob., 41, 93–107, 2004).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997
Cline, D. B. H., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stoch. Process. Appl., 49, 75–98 (1994)
Nagaev, A. V.: Integral limit theorems for large deviations when Cramér’s condition is not fulfilled I. Theory Prob. Appl., 14, 51–64 (1969a)
Nagaev, A. V.: Integral limit theorems for large deviations when Cramér’s condition is not fulfilled II. Theory Prob. Appl., 14, 193–208 (1969b)
Nagaev, A. V.: Limit theorems for large deviations when Cramér’s conditions are violated. Fiz–Mat. Nauk., 7, 17–22 (1969c) (in Russian)
Heyde, C. C.: A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitsth, 7, 303–308 (1967a)
Heyde, C. C.: On large deviation problem for sums of random variables which are not attracted to the normal law. Ann. Math. Statist., 38, 1575–1578 (1967b)
Heyde, C. C.: On large deviation probabilities in the case of attraction to a nonnormal stable law. Scankhyā A, 30, 253–258 (1968)
Nagaev, A. V.: Large deviations for sums of independent random variables, In Trans. Sixth Prague Conf. Inf. Theory Statist, Decision Functions Random Process, Academia, Prague, , 657–674, 1973
Nagaev, A. V.: Large deviations of sums of independent random variables. Ann. Prob., 7, 745–789 (1979)
Cline, D. B. H., Hsing, T.: Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails, Preprint, Texas A&M University, 1991
Klüppelberg, Mikosch: Large deviations of heavy–tailed random sums with applications in insurance and finance. J. Appl. Prob., 34, 293–308 (1997)
Mikosch, Nagaev: Large deviations of heavy–tailed sums with applications in insurance. Extremes, 1, 81–110 (1998)
Tang, Q., Su, C. Jiang, T., Zhang, J. S.: large deviations for heavy–tailed random sums in compound renewal model. Statist. Prob. Lett., 52, 91–100 (2001)
Ng, K. W., Tang, Q., Yan, J., Yang, H.: Precise large deviations for sums of random variables with consistently varying tails. J. Appl. Prob., 41, 93–107 (2004)
Hu, Y. J.: Finite Time Ruin Probabilities and Large Deviations for Generalized Compound Binomial Risk Models. Acta Mathematica Sinica, English Series, 21(5), 1099–1106 (2005)
Joag–Dev, K., Proschan, F.: Negative association of random variables with aaapplitions. Ann. Statist., 11, 286–295 (1983)
Matula, P.: A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Prob. Lett., 15, 209–213 (1992)
Su, C. Zaho, L. C., Wang, Y. B.: Moment inequalities and weak convergence for negatively associated sequences. Science in China (A), 40(2), 172–182 (1997)
Shao, Q. M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Prob., 13, 343–356 (2000)
Lehmann, E. L.: Some concepts of dependence. Ann. Math. Statist., 43, 1137–1153 (1966)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by National Science Foundation of China (No. 10271087)
Rights and permissions
About this article
Cite this article
Wang, Y.B., Wang, K.Y. & Cheng, D.Y. Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails. Acta Math Sinica 22, 1725–1734 (2006). https://doi.org/10.1007/s10114-005-0745-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-005-0745-8