Journal of Applied Probability

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

Kai W. Ng, Qihe Tang, Jia-An Yan, and Hailiang Yang

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let {Xk, 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 F̅(x) = 1 - F(x) is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums Sn and the random sums SN(t), where N(·) is a counting process independent of the sequence {Xk, 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.

Article information

J. Appl. Probab. Volume 41, Number 1 (2004), 93-107.

First available in Project Euclid: 18 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks

Consistently varying tail doubly stochastic process heavy tail Matuszewska index negative association precise large deviations random sums


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. doi:10.1239/jap/1077134670.

Export citation


  • Alam, K. and Saxena, K. M. L. (1981). Positive dependence in multivariate distributions. Commun. Statist. Theory Meth. 10, 1183--1196.
  • Bening, V. E. and Korolev, V. Yu. (2002). Generalized Poisson Models and Their Applications in Insurance and Finance. VSP, Zeist.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
  • 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.
  • Cline, D. B. H. (1994). Intermediate regular and $\Pi $ variation. Proc. London Math. Soc. 68, 594--616.
  • 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.
  • Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75--98.
  • Denuit, M., Lefèvre, C. and Utev, S. (2002). Measuring the impact of dependence between claims occurrences. Insurance Math. Econom. 30, 1--19.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • Fuk, D. H. and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Prob. Appl. 16, 660--675.
  • Gnedenko, B. V. and Korolev, V. Yu. (1996). Random Summation. Limit Theorems and Applications. CRC Press, Boca Raton, FL.
  • Grandell, J. (1976). Doubly Stochastic Poisson Processes (Lecture Notes Math. 529). Springer, Berlin.
  • Heyde, C. C. (1967a). A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitsth. 7, 303--308.
  • 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.
  • Heyde, C. C. (1968). On large deviation probabilities in the case of attraction to a nonnormal stable law. Sankhyā A 30, 253--258.
  • Jelenković, P. R. and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on--off processes. Adv. Appl. Prob. 31, 394--421.
  • Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statist. 11, 286--295.
  • 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.
  • Korolev, V. Y. (1999). On the convergence of distributions of generalized Cox processes to stable laws. Theory Prob. Appl. 43, 644--650.
  • 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.
  • 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.
  • Mikosch, T. and Nagaev, A. V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes 1, 81--110.
  • Nagaev, A. V. (1969a). Integral limit theorems for large deviations when Cramér's condition is not fulfilled. I. Theory Prob. Appl. 14, 51--64.
  • Nagaev, A. V. (1969b). Integral limit theorems for large deviations when Cramér's condition is not fulfilled. II. Theory Prob. Appl. 14, 193--208.
  • Nagaev, A. V. (1969c). Limit theorems for large deviations when Cramér's conditions are violated. Fiz-Mat. Nauk. 7, 17--22 (in Russian).
  • 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.
  • Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745--789.
  • Rozovski, L. V. (1993). Probabilities of large deviations on the whole axis. Theory Prob. Appl. 38, 53--79.
  • Schlegel, S. (1998). Ruin probabilities in perturbed risk models. Insurance Math. Econom. 22, 93--104.
  • Shao, Q. M. (2000). A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Prob. 13, 343--356.
  • 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.
  • 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.
  • 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.
  • Vinogradov, V. (1994). Refined Large Deviation Limit Theorems (Pitman Res. Notes Math. Ser. 315). Longman, Harlow.