[1] Yang Yang, Lin Jinguan,. Precise large deviation result for heavy-tailed random sumsand applications to risk theory [J]. Journal of Southeast University (English Edition), 2010, 26 (3): 498-501. [doi:10.3969/j.issn.1003-7985.2010.03.026]

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Precise large deviation result for heavy-tailed random sumsand applications to risk theory()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

26

Issue:

2010 3

Page:

498-501

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2010-09-30

Info

Title:

Precise large deviation result for heavy-tailed random sumsand applications to risk theory

Author(s):

Yang Yang¹;  2;  Lin Jinguan¹

¹ Department of Mathematics, Southeast University, Nanjing 211189, China
² School of Mathematics and Statistics, Nanjing Audit University, Nanjing 210029, China

Keywords:

precise large deviation;  random sum;  sub-exponential distribution;  renewal counting process;  customer-arrival-based insurance risk model

PACS:

O211.4

DOI:

10.3969/j.issn.1003-7985.2010.03.026

Abstract:

The differences between two sequences of nonnegative independent and identically distributed random variables with sub-exponential tails and the random index are studied. The random index is a strictly stationary renewal counting process generated by some negatively associated random variables. Using a revised large deviation result of partial sums, the elementary renewal theorem and the central limit theorem of negatively associated random variables, a precise large deviation result is derived for the random sums. The result is applied to the customer-arrival-based insurance risk model. Some uniform asymptotics for the ruin probabilities of an insurance company are obtained as the number of customers or the time tends to infinity.

References:

[1] Embrechts P, Kluppelberg C, Mikosch T. Modelling extremal events [M]. Berlin: Springer, 1997.
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[3] Mikosch T, Nagaev A V. Large deviations of heavy-tailed sums with applications in insurance [J]. Extremes, 1998, 1(1):81-110.
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[5] Tang Q, Su C, Jiang T, et al. Large deviations for heavy-tailed random sums in compound renewal model [J]. Statist and Probab Letters, 2001, 52(1):91-100.
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[9] Yang Y. The study of some topics on heavy-tailed risk models [D]. Suzhou: Department of Mathematics of Soochow University, 2008.(in Chinese)
[10] Yang Y, Wang Y. Large deviations for random variables with two-sided distributions [J]. Acta Mathematica Sinica, 2009, 52(2): 289-300.(in Chinese)
[11] Yang Y, Wang Y. The asymptotical normality of the renewal process generated by identically distributed NA random variables [J]. Journal of Lanzhou University: Natural Sciences, 2007, 43(6): 122-124.(in Chinese)
[12] Ng K W, Tang Q, Yan J, et al. Precise large deviations for the prospective-loss process [J]. J Appl Probab, 2003, 40(2): 391-400.
[13] Shen X, Lin Z, Zhang Y. Precise large deviations for the actual aggregate loss process [J]. Stochastic Analysis and Applications, 2009, 27(5): 1000-1013.

Memo

Memo:

Biographies: Yang Yang(1979—), male, doctor; Lin Jinguan(corresponding author), male, doctor, professor, jglin@seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.10671139, 11001052), the Natural Science Foundation of Jiangsu Province(No.BK2008284), China Postdoctoral Science Foundation(No.20100471365), the Natural Science Foundation of Higher Education Institutions of Jiangsu Province(No. 09KJD110003), Postdoctoral Research Program of Jiangsu Province(No.0901029C).
Citation: Yang Yang, Lin Jinguan.Precise large deviation result for heavy-tailed random sums and applications to risk theory[J].Journal of Southeast University(English Edition), 2010, 26(3):498-501.

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