Hostname: page-component-89b8bd64d-nlwjb Total loading time: 0 Render date: 2026-05-09T13:56:46.080Z Has data issue: false hasContentIssue false
  • English
  • Français

Precise large deviations for the prospective-loss process

Part of: Applications Stochastic processes Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Kai W. Ng ,
Qihe Tang ,
Jiaan Yan  and
Hailiang Yang
Kai W. Ng*
Affiliation:
University of Hong Kong
Qihe Tang*
Affiliation:
University of Amsterdam
Jiaan Yan*
Affiliation:
The 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, The 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.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we propose a customer-arrival-based insurance risk model, in which customers' potential claims are described as independent and identically distributed heavy-tailed random variables and premiums are the same for each policy. We obtain some precise large deviation results for the prospective-loss process under a mild assumption on the random index (in our case, the customer-arrival process), which is much weaker than that in the literature.

Keywords

Insurance risk modelpoint processprecise large deviationsubexponentiality

MSC classification

Primary: 60F10: Large deviations 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Secondary: 60K10: Applications (reliability, demand theory, etc.) 62P05: Applications to actuarial sciences and financial mathematics

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 391 - 400
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.CrossRefGoogle Scholar
Asmussen, S. and Klüppelberg, C. (1996). Large deviation results for subexponential tail, with applications to insurance risk. Stoch. Process. Appl. 64, 103–125.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.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
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google 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
Mikosch, T., and Nagaev, A. V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes 1, 81–110.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
Tang, Q. H., and Yan, J. A. (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.CrossRefGoogle Scholar
Tang, Q. H., 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.Google Scholar