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Precise large deviations for sums of random variables with consistently varying tails

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Kai W. Ng ,
Qihe Tang ,
Jia-An Yan  and
Hailiang Yang
Kai W. Ng*
Affiliation:
University of Hong Kong
Qihe Tang*
Affiliation:
University of Amsterdam
Jia-An Yan*
Affiliation:
Chinese Academy of Sciences, Beijing
Hailiang Yang*
Affiliation:
University of Hong Kong
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong.
∗∗∗ Postal address: Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands.
∗∗∗∗ Postal address: Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China.
∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong.
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Abstract

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ > 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N(t), where N(·) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

Keywords

Consistently varying taildoubly stochastic processheavy tailMatuszewska indexnegative associationprecise large deviationsrandom sums

MSC classification

Primary: 60F10: Large deviations 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 93 - 107
Copyright
Copyright © Applied Probability Trust 2004 

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References

Alam, K., and Saxena, K. M. L. (1981). Positive dependence in multivariate distributions. Commun. Statist. Theory Meth. 10, 1183–1196.Google Scholar
Bening, V. E. and Korolev, V. Yu. (2002). Generalized Poisson Models and Their Applications in Insurance and Finance. VSP, Zeist.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Cai, J., and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117–130.CrossRefGoogle Scholar
Cline, D. B. H. (1994). Intermediate regular and Π variation. Proc. London Math. Soc. 68, 594–616.Google Scholar
Cline, D. B. H., and Hsing, T. (1991). Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint, Texas A&M University.Google Scholar
Cline, D. B. H., and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.Google Scholar
Denuit, M., Lefèvre, C., and Utev, S. (2002). Measuring the impact of dependence between claims occurrences. Insurance Math. Econom. 30, 1–19.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Fuk, D. H., and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Prob. Appl. 16, 660–675.Google Scholar
Gnedenko, B. V. and Korolev, V. Yu. (1996). Random Summation. Limit Theorems and Applications. CRC Press, Boca Raton, FL.Google Scholar
Grandell, J. (1976). Doubly Stochastic Poisson Processes (Lecture Notes Math. 529). Springer, Berlin.Google Scholar
Heyde, C. C. (1967a). A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitsth. 7, 303–308.Google Scholar
Heyde, C. C. (1967b). On large deviation problems for sums of random variables which are not attracted to the normal law. Ann. Math. Statist. 38, 1575–1578.Google Scholar
Heyde, C. C. (1968). On large deviation probabilities in the case of attraction to a nonnormal stable law. Sankhyā A 30, 253–258.Google Scholar
Jelenković, P. R., and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on–off processes. Adv. Appl. Prob. 31, 394–421.Google Scholar
Joag-Dev, K., and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statist. 11, 286–295.CrossRefGoogle Scholar
Klüppelberg, C., and Mikosch, T. (1997). Large deviations of heavy-tailed random sums with applications in insurance and finance. J. Appl. Prob. 34, 293–308.Google Scholar
Korolev, V. Y. (1999). On the convergence of distributions of generalized Cox processes to stable laws. Theory Prob. Appl. 43, 644–650.CrossRefGoogle Scholar
Korolev, V. Y. (2001). Asymptotic properties of the extrema of generalized Cox processes and their application to some problems in financial mathematics. Theory Prob. Appl. 45, 136–147.CrossRefGoogle Scholar
Meerschaert, M. M., and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice. John Wiley, New York.Google Scholar
Mikosch, T., and Nagaev, A. V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes 1, 81–110.Google Scholar
Nagaev, A. V. (1969a). Integral limit theorems for large deviations when Cramér's condition is not fulfilled. I. Theory Prob. Appl. 14, 51–64.CrossRefGoogle Scholar
Nagaev, A. V. (1969b). Integral limit theorems for large deviations when Cramér's condition is not fulfilled. II. Theory Prob. Appl. 14, 193–208.Google Scholar
Nagaev, A. V. (1969c). Limit theorems for large deviations when Cramér's conditions are violated. Fiz-Mat. Nauk. 7, 17–22 (in Russian).Google Scholar
Nagaev, S. V. (1973). Large deviations for sums of independent random variables. In Trans. Sixth Prague Conf. Inf. Theory Statist. Decision Functions Random Processes, Academia, Prague, pp. 657–674.Google Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.Google Scholar
Rozovski, L. V. (1993). Probabilities of large deviations on the whole axis. Theory Prob. Appl. 38, 53–79.CrossRefGoogle Scholar
Schlegel, S. (1998). Ruin probabilities in perturbed risk models. Insurance Math. Econom. 22, 93–104.Google Scholar
Shao, Q. M. (2000). A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Prob. 13, 343–356.Google Scholar
Tang, Q., and Tsitsiashvili, G. (2004). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108, 299–325.Google Scholar
Tang, Q., and Yan, J. (2002). A sharp inequality for the tail probabilities of sums of i.i.d. r.v.'s with dominatedly varying tails. Sci. China A 45, 1006–1011.Google Scholar
Tang, Q., Su, C., Jiang, T., and Zhang, J. S. (2001). Large deviations for heavy-tailed random sums in compound renewal model. Statist. Prob. Lett. 52, 91–100.CrossRefGoogle Scholar
Vinogradov, V. (1994). Refined Large Deviation Limit Theorems (Pitman Res. Notes Math. Ser. 315). Longman, Harlow.Google Scholar