##
**Extended precise large deviations of random sums in the presence of END structure and consistent variation.**
*(English)*
Zbl 1236.60031

Summary: The study of precise large deviations of random sums is an important topic in insurance and finance. In this paper, extended precise large deviations of random sums in the presence of an extended negatively de-
pendent (END) structure and consistent variation are investigated. The obtained results extend those of Y. Chen and W. Zhang [Stat. Probab. Lett. 77, No. 5, 530–538 (2007; Zbl 1117.60025)] and Y. Chen, A. Chen and K. W. Ng [J. Appl. Probab. 47, No. 4, 908–922 (2010; Zbl 1213.60058)]. As an application, precise large deviations of the prospective-loss process of a quasirenewal model are considered.

### MSC:

60F10 | Large deviations |

### Keywords:

precise large deviations of random sums; insurance; finance; prospective-loss process of a quasirenewal model
PDFBibTeX
XMLCite

\textit{S. Wang} and \textit{W. Wang}, J. Appl. Math. 2012, Article ID 436531, 12 p. (2012; Zbl 1236.60031)

Full Text:
DOI

### References:

[1] | D. B. H. Cline and T. H. Samorodnitsky, “Subexponentiality of the product of independent random variables,” Stochastic Processes and Their Applications, vol. 49, no. 1, pp. 75-98, 1994. · Zbl 0799.60015 · doi:10.1016/0304-4149(94)90113-9 |

[2] | K. W. Ng, Q. H. Tang, J.-A. Yan, and H. L. Yang, “Precise large deviations for sums of random variables with consistently varying tails,” Journal of Applied Probability, vol. 41, no. 1, pp. 93-107, 2004. · Zbl 1051.60032 · doi:10.1239/jap/1077134670 |

[3] | S. J. Wang and W. S. Wang, “Precise large deviations for sums of random variables with consistently varying tails in multi-risk models,” Journal of Applied Probability, vol. 44, no. 4, pp. 889-900, 2007. · Zbl 1134.60322 · doi:10.1239/jap/1197908812 |

[4] | C. Klüppelberg and T. Mikosch, “Large deviations of heavy-tailed random sums with applications in insurance and finance,” Journal of Applied Probability, vol. 34, no. 2, pp. 293-308, 1997. · Zbl 1140.60313 · doi:10.1016/j.spl.2007.09.040 |

[5] | K. W. Ng, Q. H. Tang, J.-A. Yan, and H. L. Yang, “Precise large deviations for the prospective-loss process,” Journal of Applied Probability, vol. 40, no. 2, pp. 391-400, 2003. · Zbl 1028.60024 · doi:10.1239/jap/1053003551 |

[6] | Y. Liu, “Precise large deviations for negatively associated random variables with consistently varying tails,” Statistics & Probability Letters, vol. 77, no. 2, pp. 181-189, 2007. · Zbl 1111.60017 · doi:10.1016/j.spl.2006.07.002 |

[7] | Y. Chen and W. P. Zhang, “Large deviations for random sums of negatively dependent random variables with consistently varying tails,” Statistics & Probability Letters, vol. 77, no. 5, pp. 530-538, 2007. · Zbl 1117.60025 · doi:10.1016/j.spl.2006.08.021 |

[8] | J. Lin, “The general principle for negatively associated random variables with consistently varying tails,” Statistics & Probability Letters, vol. 78, no. 6, pp. 181-189, 2008. |

[9] | A. Baltr\Bunas, R. Leipus, and J. \vSiaulys, “Precise large deviation results for the total claim amount under subexponential claim sizes,” Statistics & Probability Letters, vol. 78, no. 10, pp. 1206-1214, 2008. · Zbl 1145.60018 · doi:10.1016/j.spl.2007.11.016 |

[10] | X. Shen and Z. Lin, “Precise large deviations for randomly weighted sums of negatively dependent random variables with consistently varying tails,” Statistics & Probability Letters, vol. 78, no. 18, pp. 3222-3229, 2008. · Zbl 1154.60316 · doi:10.1016/j.spl.2008.06.007 |

[11] | Y. Q. Chen, K. C. Yuen, and K. W. Ng, “Precise large deviations of random sums in the presence of negatively dependence and consistent variation,” Methodology and Computing in Applied Probability, vol. 13, no. 4, pp. 821-833, 2011. · Zbl 1213.60058 · doi:10.1239/jap/1294170508 |

[12] | L. Liu, “Precise large deviations for dependent random variables with heavy tails,” Statistics & Probability Letters, vol. 79, no. 9, pp. 1290-1298, 2009. · Zbl 1163.60012 · doi:10.1016/j.spl.2009.02.001 |

[13] | Q. H. 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 and Their Applications, vol. 108, no. 2, pp. 299-325, 2003. · Zbl 1075.91563 · doi:10.1016/j.spa.2003.07.001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.