Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-25T05:43:39.268Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted sums of subexponential random variables and their maxima

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Yiqing Chen ,
Kai W. Ng  and
Qihe Tang
Yiqing Chen*
Affiliation:
Guangdong University of Technology
Kai W. Ng*
Affiliation:
The University of Hong Kong
Qihe Tang*
Affiliation:
Concordia University
*
∗ Postal address: School of Economics and Management, Guangdong University of Technology, East Dongfeng Road 729, Guangzhou 510090, Guangdong, P. R. China. Email address: yiqch@263.net
∗∗ Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: kaing@hku.hk
∗∗∗ Postal address: Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, H4B 1R6, Canada. Email address: qtang@mathstat.concordia.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {Xk, k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {wk, k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1nwkXk and the maximum of weighted sums max1≤m≤n∑k=1mwkXk, subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss Xk during the kth year, with a discount or inflation factor wk.

Keywords

AsymptoticsMatuszewska indexweighted summaximumsubexponentialitytail probabilityuniformityruin probabilitydiscounted loss

MSC classification

Secondary: 60G50: Sums of independent random variables; random walks 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 510 - 522
Copyright
Copyright © Applied Probability Trust 2005 

References

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Cline, D. B. H. (1983). Infinite series of random variables with regularly varying tails. Tech. Rep. 83-24, Institute of Applied Mathematics and Statistics, University of British Columbia.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.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
Davis, R. and Resnick, S. (1988). Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch. Process. Appl. 30, 41–68.Google Scholar
De Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes (Math. Centre Tracts 32). Mathematisch Centrum, Amsterdam.Google Scholar
Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243–256.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Konstantinides, D., Tang, Q. and Tsitsiashvili, G. (2002). Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econom. 31, 447–460.CrossRefGoogle Scholar
Korshunov, D. A. (2001). Large deviation probabilities for the maxima of sums of independent summands with a negative mean and a subexponential distribution. Teor. Veroyatnost. i Primenen. 46, 387–397 (in Russian). English translation: Theory Prob. Appl. 46 (2002), 355-366.Google Scholar
Ng, K. W., Tang, Q. and Yang, H. (2002). Maxima of sums of heavy-tailed random variables. Astin Bull. 32, 43–55.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Tang, Q. (2004). The ruin probability of a discrete time risk model under constant interest rate with heavy tails. Scand. Actuarial J., 229–240.CrossRefGoogle Scholar
Tang, Q. and Tsitsiashvili, G. (2003). 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
Zerner, M. P. W. (2002). Integrability of infinite weighted sums of heavy-tailed i.i.d. random variables. Stoch. Process. Appl. 99, 81–94.Google Scholar