Precise large deviations of aggregate claims in a nonstandard risk model with arbitrary dependence between claim sizes and waiting times

Qingwu Gao; Wenlei Pan; Qingwu Gao; Wenlei Pan

doi:10.3934/math.2023113

AIMS Mathematics

2023, Volume 8, Issue 1: 2191-2200. doi: 10.3934/math.2023113

Previous Article Next Article

Research article

Precise large deviations of aggregate claims in a nonstandard risk model with arbitrary dependence between claim sizes and waiting times

Qingwu Gao ^,,
Wenlei Pan

School of Statistics and Data Science, Nanjing Audit University, Nanjing, China

Received: 01 August 2022 Revised: 30 September 2022 Accepted: 09 October 2022 Published: 31 October 2022
MSC : Primary 60F10; Secondary 62P05, 91B30

Recently, Chen et al.^[3] investigated the precise large deviations of aggregate claims in a renewal risk model with arbitrary dependence between claim sizes and their waiting times. In this paper, we extend their results to a nonstandard risk model in which various dependence structures are imposed on the modeling components, and obtain the asymptotic lower and upper bounds of precise large deviations for aggregate claims, which hold uniformly for all $ x $ in a $ t $-interval.
- asymptotics,
- precise large deviations,
- dependence,
- consistent variation,
- aggregate claims
Citation: Qingwu Gao, Wenlei Pan. Precise large deviations of aggregate claims in a nonstandard risk model with arbitrary dependence between claim sizes and waiting times[J]. AIMS Mathematics, 2023, 8(1): 2191-2200. doi: 10.3934/math.2023113

Related Papers:

Abstract

Recently, Chen et al.^[3] investigated the precise large deviations of aggregate claims in a renewal risk model with arbitrary dependence between claim sizes and their waiting times. In this paper, we extend their results to a nonstandard risk model in which various dependence structures are imposed on the modeling components, and obtain the asymptotic lower and upper bounds of precise large deviations for aggregate claims, which hold uniformly for all $ x $ in a $ t $-interval.

References

[1]	N. Bingham, C. Goldie, J. Teugels, Regular variation, Cambridge: Cambridge University Press, 1987. http://dx.doi.org/10.1017/CBO9780511721434
[2]	Y. Chen, K. Yuen, Precise large deviations of aggregate claims in a size-dependent renewal risk model, Insur. Math. Econ., 51 (2012), 457–461. http://dx.doi.org/10.1016/j.insmatheco.2012.06.010 doi: 10.1016/j.insmatheco.2012.06.010
[3]	Y. Chen, T. White, K. Yuen, Precise large deviations of aggregate claims with arbitrary dependence between claim sizes and waiting times, Insur. Math. Econ., 97 (2021), 1–6. http://dx.doi.org/10.1016/j.insmatheco.2020.12.003 doi: 10.1016/j.insmatheco.2020.12.003
[4]	Y. Chen, K. Yuen, K. Ng, Precise large deviations of random sums in presence of negative dependence and consistent variation, Methodol. Comput. Appl. Probab., 13 (2011), 821–833. http://dx.doi.org/10.1007/s11009-010-9194-7 doi: 10.1007/s11009-010-9194-7
[5]	D. Cline, T. Hsing, Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails, Texas A & M University, in press.
[6]	P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling extremal events: for insurance and finance, Berlin: Springer, 1997. http://dx.doi.org/10.1007/978-3-642-33483-2
[7]	Q. Gao, X. Liu, C. Chai, Asymptotic bounds for precise large deviations in a compound risk model under dependence structures, J. Math. Inequal., 14 (2020), 1067–1082. http://dx.doi.org/10.7153/jmi-2020-14-69 doi: 10.7153/jmi-2020-14-69
[8]	W. He, D. Cheng, Y. Wang, Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables, Stat. Probabil. Lett., 83 (2013), 331–338. http://dx.doi.org/10.1016/j.spl.2012.09.019 doi: 10.1016/j.spl.2012.09.019
[9]	T. Jiang, S. Cui, R. Ming, Large deviations for the stochastic present value of aggregate claims in the renewal risk model, Stat. Probabil. Lett., 101 (2015), 83–91. http://dx.doi.org/10.1016/j.spl.2015.02.020 doi: 10.1016/j.spl.2015.02.020
[10]	R. Kaas, Q. Tang, A large deviation result for aggregate claims with dependent claim occurencies, Insur. Math. Econ., 36 (2005), 251–259. http://dx.doi.org/10.1016/j.insmatheco.2005.01.004 doi: 10.1016/j.insmatheco.2005.01.004
[11]	D. Konstantinides, F. Loukissas, Precise large deviations for sums of negatively dependent random variables with common long-tailed distributions, Commun. Stat.-Theor. M., 40 (2011), 3663–3671. http://dx.doi.org/10.1080/03610926.2011.581186 doi: 10.1080/03610926.2011.581186
[12]	C. Kl$\rm\ddot{u}$ppelberg, T. Mikosch, Large deviations of heavy-tailed random sums with applications in insurance and finance, J. Appl. Probab., 34 (1997), 293–308. http://dx.doi.org/10.2307/3215371 doi: 10.2307/3215371
[13]	J. Li, Q. Tang, R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Adv. Appl. Probab., 42 (2010), 1126–1146. http://dx.doi.org/10.1239/aap/1293113154 doi: 10.1239/aap/1293113154
[14]	L. Liu, Precise large deviations for dependent variables with heavy tails, Stat. Probabil. Lett., 79 (2009), 1290–1298. http://dx.doi.org/10.1016/j.spl.2009.02.001 doi: 10.1016/j.spl.2009.02.001
[15]	X. Liu, Q. Gao, Y. Wang, A note on a dependent risk model with constant interest rate, Stat. Probabil. Lett., 82 (2012), 707–712. http://dx.doi.org/10.1016/j.spl.2011.12.016 doi: 10.1016/j.spl.2011.12.016
[16]	X. Liu, C. Yu, Q. Gao, Precise large deviations of aggregate claim amount in a dependent renewal risk model, Commun. Stat.-Theor. M., 46 (2017), 2354–2363. http://dx.doi.org/10.1080/03610926.2015.1044666 doi: 10.1080/03610926.2015.1044666
[17]	F. Loukissas, Precise large deviations for long-tailed distributions, J. Theor. Probab., 25 (2012), 913–924. http://dx.doi.org/10.1007/s10959-011-0367-2 doi: 10.1007/s10959-011-0367-2
[18]	D. Lu, L. Song, Y. Xu, Precise large deviations for sums of independent random variables with consistently varying tails, Commun. Stat.-Theor. M., 43 (2014), 28–43. http://dx.doi.org/10.1080/03610926.2011.654041 doi: 10.1080/03610926.2011.654041
[19]	T. Mikosch, A. Nagaev, Large deviations of heavy-tailed sums with applications in insurance, Extremes, 1 (1998), 81–110. http://dx.doi.org/10.1023/A:1009913901219 doi: 10.1023/A:1009913901219
[20]	K. Ng, Q. Tang, J. Yan, H. Yang, Precise large deviations for sums of random variables with consistently varying tails, J. Appl. Probab., 41 (2004), 93–107. http://dx.doi.org/10.1239/jap/1077134670 doi: 10.1239/jap/1077134670
[21]	T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, Stochastic processes for insurance and finance, New York: Wiley, 1999. http://dx.doi.org/10.1002/9780470317044
[22]	X. Shen, M. Xu, E. Atta Mills, Precise large deviation results for sums of sub-exponential claims in a size-dependent renewal risk model, Stat. Probabil. Lett., 114 (2016), 6–13. http://dx.doi.org/10.1016/j.spl.2016.03.002 doi: 10.1016/j.spl.2016.03.002
[23]	Q. Tang, Insensitivity to negative dependence of the asymptotic behavior of precise large deviations, Electron. J. Probab., 11 (2006), 107–120. http://dx.doi.org/10.1214/EJP.v11-304 doi: 10.1214/EJP.v11-304
[24]	Q. Tang, C. Su, T. Jiang, J. Zhang, Large deviations for heavy-tailed random sums in compound renewal model, Stat. Probabil. Lett., 52 (2001), 91–100. http://dx.doi.org/10.1016/S0167-7152(00)00231-5 doi: 10.1016/S0167-7152(00)00231-5
[25]	K. Wang, Y. Wang, Q. Gao, Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Proba., 15 (2013), 109–124. http://dx.doi.org/10.1007/s11009-011-9226-y doi: 10.1007/s11009-011-9226-y
[26]	K. Wang, L. Chen, Precise large deviations for the aggregate claims in a dependent compound renewal risk model, J. Inequal. Appl., 2019 (2019), 257. http://dx.doi.org/10.1186/s13660-019-2209-1 doi: 10.1186/s13660-019-2209-1
[27]	S. Wang, W. Wang, Precise large deviations for sums of random variables with consistently varying tails in multi-risk models, J. Appl. Probab., 44 (2007), 889–900. http://dx.doi.org/10.1239/jap/1197908812 doi: 10.1239/jap/1197908812
[28]	Y. Wang, D. Cheng, Basic renewal theorems for a random walks with widely dependent increments, J. Math. Anal. Appl., 384 (2011), 597–606. http://dx.doi.org/10.1016/j.jmaa.2011.06.010 doi: 10.1016/j.jmaa.2011.06.010
[29]	Y. Yang, K. Wang, Precise large deviations for dependent random variables with applications to the compound renewal risk model, Rocky Mountain J. Math., 43 (2013), 1395–1414. http://dx.doi.org/1216/RMJ-2013-43-4-1395

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)