Abstract
We consider a compound renewal risk model with individual claim sizes and interarrival times forming a sequence of independent identically distributed nonnegative random pairs with a generic pair (X, θ) and the numbers of claims caused by individual events constituting another sequence of independent identically distributed positive integer-valued random variables, independent of the random pairs. Allowing X and θ be arbitrarily dependent, we derive precise large deviations for aggregate claims of such a compound renewal risk model in the presence of dominatedly varying claim sizes, which extends some known results in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Aleškevičienė, R. Leipus, and J. Šiaulys, Tail behavior of random sums under consistent variation with application to the compound renewal risk model, Extremes, 11:261–279, 2008.
A. Baltrūnas, R. Leipus, and J. Šiaulys, Precise large deviation results for the total claim amount under subexponential claim sizes, Stat. Probab. Lett., 78(10):1206–1214, 2008.
X. Bi and S. Zhang, Precise large deviations of aggregate claims in a risk model with regression-type size-dependence, Stat. Probab. Lett., 83(10):2248–2255, 2013.
Y. Chen, T. White, and K.C. Yuen, Precise large deviations of aggregate claims with arbitrary dependence between claims sizes and waiting times, Insur. Math. Econ., 97:1–6, 2021.
Y. Chen and K.C. Yuen, Precise large deviations of aggregate claims in a size-dependent renewal risk model, Insur. Math. Econ., 51(2):457–461, 2012.
Y. Chen, K.C. Yuen, and K.W. Ng, Precise large deviations of random sums in presence of negative dependence and consistent variation, Methodol. Comput. Appl. Probab., 13(4):821–833, 2011.
P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, Heidelberg, 1997.
S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, Springer, New York, 2013.
H. Guo, S. Wang, and C. Zhang, Precise large deviations of aggregate claims in a compound size-dependent renewal risk model, Commun. Stat., Theory Methods, 46(3):1107–1116, 2017.
R. Kass and Q. Tang, A large deviation result for aggregate claims with dependent claim occurrences, Insur. Math. Econ., 36(3):251–259, 2005.
C. Klüppelberg and T. Mikosch, Large deviations of heavy-tailed random sums with applications in insurance and finance, J. Appl. Probab., 34(2):293–308, 1997.
D. Konstantinides and F. Loukissas, Precise large deviations for consistently varying-tailed distributions in the compound renewal risk model, Lit. Math. J., 50(4):391–400, 2010.
L. Liu, Precise large deviations for dependent random variables with heavy tails, Stat. Probab. Lett., 79(9):1290–1298, 2009.
Y. Liu, Precise large deviations for negatively associated random variables with consistently varying tails, Stat. Probab. Lett., 77(2):181–189, 2007.
F. Loukissas, Precise large deviations for long-tailed distributions, J. Theor. Probab., 25:913–924, 2012.
T. Mikosch and O. Wintenberger, Precise large deviations for dependent regularly varying sequences, Probab. Theory Relat. Fields, 156(3-4):851–887, 2013.
K.W. Ng, Q. Tang, J. Jian, and H. Yang, Precise large deviations for the prospective-loss process, J. Appl. Probab., 40(2):391–400, 2003.
K.W. Ng, Q. Tang, J.-A. Yan, and H. Yang, Precise large deviations for sums of random variables with consistently varying tails, J. Appl. Probab., 41(1):93–107, 2004.
Q. Tang, C. Su, T. Jiang, and J. Zhang, Large deviations for heavy-tailed random sums in compound renewal model, Stat. Probab. Lett., 52(1):91–100, 2001.
Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Processes Appl., 108(2):299–325, 2003.
Q. Tang and G. Tsitsiashvili, Insensitivity to negative dependence of the asymptotic behavior of precise large deviations, Electron. J. Probab., 11:107–120, 2006.
Y. Yang, R. Leipus, and J. Šiaulys, Precise large deviations for compound random sums in the presence of dependence structures, Comput. Math. Appl., 64(6):2074–2083, 2012.
Y. Yang, R. Leipus, and J. Šiaulys, Precise large deviations for actual aggregate loss process in a dependent compound customer-arrival-based insurance risk model, Lith. Math. J., 53(4):448–470, 2013.
Y. Yang and L. Sha, Precise large deviations for aggregate claims, Commun. Stat., Theory Methods, 45(10):2801–2809, 2016.
Y. Yang and K. Wang, Precise large deviations for dependent random variables with applications to the compound renewal risk model, Rocky Mt. J. Math., 43(4):1395–1414, 2013.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
* This work is supported by Anhui University Center for Pure Mathematics and the Provincial Natural Science Research Project of Anhui Colleges (KJ2021A0060).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, S., Gao, Y. Precise large deviations for aggregate claims of a compound renewal risk model with arbitrary dependence between claim sizes and waiting times*. Lith Math J 62, 542–552 (2022). https://doi.org/10.1007/s10986-022-09581-w
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-022-09581-w