Abstract
Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sums \(\sum\limits_{i = 1}^n {\theta _i X_i }\) and their maxima. In this paper, we generalize their work to the situation that {X i , i ≥ 1} is a sequence of upper tail asymptotically independent random variables with common distribution from the class \(\mathcal{D} \cap \mathcal{L}\), and {θ i , i ≥ 1} is a sequence of nonnegative random variables, independent of {X i , i ≥ 1} and satisfying some regular conditions. Moreover, no additional assumption is required on the dependence structure of {θ i , i ≥ 1}.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bingham, N.H., Goldie, C.M., Teugels, J.L. Regular Variation. Cambridge University Press, Cambridge, 1987
Chen, Y.Q., Ng, K., Tang, Q.H. Weighted sums of subexponential random variables and their maxima. Advances in Applied Probability, 37(2): 510–522 (2005)
Cline, D.B.H., Samorodnitsky, G. Subexponentiality of the product of independent random variables. Stochastic Processes and Their Applications, 49: 75–98 (1994)
Gao, Q.W., Wang, Y.B. Randomly weighted sums with dominated varying-tailed increments and application to risk theory. Journal of the Korean Statistical Society, 39: 305–314 (2010)
Geluk, J., Tang Q.H. Asymptotic tail probabilities of sums of dependent subexponential random variables. Journal of Theoretical Probability, 22(4): 871–882 (2009)
Joe, H. Multivariate Models and Dependence Concepts. Chapman and Hall, London, 1997
Nelsen, R.B. An Introduction to Copulas. 2nd, Springer-Verlag, New York, 2006
Shen, X.M., Lin, Z.Y., Zhang Y. Uniform estimate for maximum of randomly weighted sums with applications to ruin theory. Methodology and Computing in Applied Probability, 11(4): 669–685 (2009)
Tang, Q.H., Tsitsiashvili G. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Processes and Their Applications, 108(2): 299–325 (2003)
Weng, C.G., Zhang, Y., Tan, K.S. Ruin probabilities in a discrete time risk model with dependent risks of heavy tail. Scandinavian Actuarial Journal, 2009(3): 205–218 (2009)
Wang, D.C., Su, C., Zeng, Y. Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory. Science in China (Series A: Mathematics), 48(10): 1379–1394 (2005)
Wang, Y.F., Yin, C.C. Approximation for the ruin probabilities in a discrete time risk model with dependent risks. Statistics and Probability Letters, 80: 1335–1342 (2010)
Zhang, Y., Shen, X.M., Weng, C.G. Approximation of the tail probability of randomly weighted sums and applications. Stochastic Processes and Their Applications, 119: 655–675 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (No. 11071045, No. 11171179, No. 11201080, No. 11301391) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20133705110002).
Rights and permissions
About this article
Cite this article
Wang, Yf., Yin, Cc. & Zhang, Xs. Uniform estimate for the tail probabilities of randomly weighted sums. Acta Math. Appl. Sin. Engl. Ser. 30, 1063–1072 (2014). https://doi.org/10.1007/s10255-014-0446-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-014-0446-0
Keywords
- uniform estimate
- randomly weighted sums
- upper tail asymptotically independence
- class \(\mathcal{D} \cap \mathcal{L}\)