## Electronic Journal of Probability

### Insensitivity to Negative Dependence of the Asymptotic Behavior of Precise Large Deviations

Qihe Tang

#### Abstract

Since the pioneering works of C.C. Heyde, A.V. Nagaev, and S.V. Nagaev in 1960's and 1970's, the precise asymptotic behavior of large-deviation probabilities of sums of heavy-tailed random variables has been extensively investigated by many people, but mostly it is assumed that the random variables under discussion are independent. In this paper, we extend the study to the case of negatively dependent random variables and we find out that the asymptotic behavior of precise large deviations is insensitive to the negative dependence.

#### Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 4, 107-120.

Dates
Accepted: 11 February 2006
First available in Project Euclid: 31 May 2016

https://projecteuclid.org/euclid.ejp/1464730539

Digital Object Identifier
doi:10.1214/EJP.v11-304

Mathematical Reviews number (MathSciNet)
MR2217811

Zentralblatt MATH identifier
1109.60021

Subjects
Primary: 60F10: Large deviations
Secondary: 60E15: Inequalities; stochastic orderings

Rights

#### Citation

Tang, Qihe. Insensitivity to Negative Dependence of the Asymptotic Behavior of Precise Large Deviations. Electron. J. Probab. 11 (2006), paper no. 4, 107--120. doi:10.1214/EJP.v11-304. https://projecteuclid.org/euclid.ejp/1464730539

#### References

• Alam, Khursheed; Saxena, K. M. Lal. Positive dependence in multivariate distributions. Comm. Statist. A–-Theory Methods 10 (1981), no. 12, 1183–1196.
• Baltrunas, Aleksandras; Klüppelberg, Claudia. Subexponential distributions–-large deviations with applications to insurance and queueing models. Festschrift in honour of Daryl Daley. Aust. N. Z. J. Stat. 46 (2004), no. 1, 145–154.
• Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2
• Bingham, N. H.; Nili Sani, H. R. Summability methods and negatively associated random variables. Stochastic methods and their applications. J. Appl. Probab. 41A (2004), 231–238.
• Block, Henry W.; Savits, Thomas H.; Shaked, Moshe. Some concepts of negative dependence. Ann. Probab. 10 (1982), no. 3, 765–772.
• Cai, Jun; Tang, Qihe. On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Probab. 41 (2004), no. 1, 117–130.
• Cline, Daren B. H. Intermediate regular and $\Pi$ variation. Proc. London Math. Soc. (3) 68 (1994), no. 3, 594–616.
• Cline, Daren B. H. ; T. Hsing. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint (1991), Texas A&amp;M; University.
• Cline, D. B. H.; Samorodnitsky, G. Subexponentiality of the product of independent random variables. Stochastic Process. Appl. 49 (1994), no. 1, 75–98.
• Ebrahimi, Nader; Ghosh, Malay. Multivariate negative dependence. Comm. Statist. A–-Theory Methods 10 (1981), no. 4, 307–337.
• Fuk, D. H.; Nagaev, S. V. Probabilistic inequalities for sums of independent random variables. (Russian) Teor. Verojatnost. i Primenen. 16 (1971), 660–675. (45 #2772)
• Heyde, C. C. A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 1967 303–308. (35 #7380)
• Heyde, C. C. On large deviation problems for sums of random variables which are not attracted to the normal law. Ann. Math. Statist. 38 1967 1575–1578. (36 #4616)
• Heyde, C. C. On large deviation probabilities in the case of attraction to a non-normal stable law. Sankhyā Ser. A 30 1968 253–258. (39 #2199)
• Joag-Dev, Kumar; Proschan, Frank. Negative association of random variables, with applications. Ann. Statist. 11 (1983), no. 1, 286–295.
• Kotz, Samuel; Balakrishnan, N.; Johnson, Norman L.. Continuous multivariate distributions. Vol. 1. Models and applications. Second edition. Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience, New York, 2000. xxii+722 pp. ISBN: 0-471-18387-3
• Lehmann, E. L. Some concepts of dependence. Ann. Math. Statist. 37 1966 1137–1153. (34 #2101)
• Mikosch, T.; Nagaev, A. V. Large deviations of heavy-tailed sums with applications in insurance. Extremes 1 (1998), no. 1, 81–110.
• Nagaev, A. V. Integral limit theorems with regard to large deviations when Cramér's condition is not satisfied. I. (Russian) Teor. Verojatnost. i Primenen. 14 1969 51–63. (40 #915a)
• Nagaev, A. V. Integral limit theorems with regard to large deviations when Cramer's condition is not satisfied. II. Theory Probab. Appl. 14 (1969b), 193–208.
• Nagaev, A. V. Limit theorems that take into account large deviations when Cramér's condition is violated. (Russian) Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 13 1969 no. 6, 17–22. (43 #8108)
• Nagaev, S. V. Large deviations for sums of independent random variables. Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Tech. Univ., Prague, 1971; dedicated to the memory of Antonín Špaček), pp. 657–674. Academia, Prague, 1973. (50 #14901)
• Nagaev, S. V. Large deviations of sums of independent random variables. Ann. Probab. 7 (1979), no. 5, 745–789.
• Ng, Kai W.; Tang, Qihe; Yan, Jia-An; Yang, Hailiang. Precise large deviations for sums of random variables with consistently varying tails. J. Appl. Probab. 41 (2004), no. 1, 93–107.
• Pinelis, I. F. Asymptotic equivalence of the probabilities of large deviations for sums and maximum of independent random variables. (Russian) Limit theorems of probability theory, 144–173, 176, Trudy Inst. Mat., 5, "Nauka" Sibirsk. Otdel., Novosibirsk, 1985.
• Rozovskiĭ, L. V. Probabilities of large deviations on the whole axis. (Russian) Teor. Veroyatnost. i Primenen. 38 (1993), no. 1, 79–109; translation in Theory Probab. Appl. 38 (1993), no. 1, 53–79
• Stadtmüller, U.; Trautner, R. Tauberian theorems for Laplace transforms. J. Reine Angew. Math. 311/312 (1979), 283–290.
• Tang, Qihe; Su, Chun; Jiang, Tao; Zhang, Jinsong. Large deviations for heavy-tailed random sums in compound renewal model. Statist. Probab. Lett. 52 (2001), no. 1, 91–100.
• Tang, Qihe; Tsitsiashvili, Gurami. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process. Appl. 108 (2003), no. 2, 299–325.
• Tang, Qihe; Yan, Jia'an. A sharp inequality for the tail probabilities of sums of i.i.d. r.v.'s with dominatedly varying tails. Sci. China Ser. A 45 (2002), no. 8, 1006–1011.
• Vinogradov, Vladimir. Refined large deviation limit theorems. Pitman Research Notes in Mathematics Series, 315. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. xii+212 pp. ISBN: 0-582-25499-X