Kaas, Rob and Tang, Qihe (2004) *Introducing a Dependence Structure to the Occurences in Studying Precise Large Deviations for the Total Claim Amount.* Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.

| PDF - Published Version 227Kb |

## Abstract

In this paper we study precise large deviations for a compound sum of claims, in which the claims arrive in groups and the claim numbers in the groups may follow a

certain negative dependence structure. We try to build a platform both for the classical large deviation theory and for the modern stochastic ordering theory.

Divisions: | Concordia University > Faculty of Arts and Science > Mathematics and Statistics |
---|---|

Item Type: | Monograph (Technical Report) |

Authors: | Kaas, Rob and Tang, Qihe |

Series Name: | Department of Mathematics & Statistics. Technical Report No. 13/04 |

Corporate Authors: | Concordia University. Department of Mathematics & Statistics |

Institution: | Concordia University |

Date: | 12 December 2004 |

Keywords: | Consistent variation; Matuszewska index; Negative cumulative dependence; Precise large deviations; Random sums; Stop-loss order. |

ID Code: | 6662 |

Deposited By: | DIANE MICHAUD |

Deposited On: | 02 Jun 2010 12:27 |

Last Modified: | 08 Dec 2010 18:24 |

References: | Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular Variation. Cambridge University Press, Cambridge, 1987.
Cai, J.; Tang, Q. On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. Journal of Applied Probability 41 (2004), no. 1, 117-130. Cline, D. B. H. Intermediate regular and ∏ variation. Proceedings of the London Mathematical Society (3rd Series) 68 (1994), no. 3, 594{616. Cline, D. B. H.; Hsing, T. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint(1991), Texas A & M University. Cline, D. B. H.; Samorodnitsky, G. Subexponentiality of the product of independent random variables. Stochastic Processes and their Applications 49 (1994), no. 1, 75-98. Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D. The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics & Economics 31 (2002a), no. 1, 3-33. Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D. The concept of comonotonicity in actuarial science and finance: applications. Insurance: Mathematics & Economics 31 (2002b), no. 2, 133-161. Dhaene, J.; Goovaerts, M. J. Dependency of risks and stop-loss order. Astin Bulletin 26 (1996), no. 2, 201-212. Denuit, M.; Dhaene, J.; Ribas, C. Does positive dependence between individual risks increase stop-loss premiums? Insurance: Mathematics & Economics 28 (2001), no. 3, 305-308. Denuit, M.; Lefµevre, C.; Utev, S. Measuring the impact of dependence between claims occurrences. Insurance: Mathematics & Economics 30 (2002), no. 1, 1-19. Embrechts, P.; Klüppelberg, C.; Mikosch, T. Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin, 1997. Fuk, D. H.; Nagaev, S. V. Probabilistic inequalities for sums of independent random variables. Theory of Probability and its Applications 16 (1971), 643-660. Jelenkovic, P. R.; Lazar, A. A. Asymptotic results for multiplexing subexponential on-off processes. Advances in Applied Probability 31 (1999), no. 2, 394-421. Kaas, R.; Goovaerts, M. J.; Tang, Q. Some useful counterexamples regarding comonotonicity. Belgian Actuarial Bulletin 4 (2004), 1-4. Klüppelberg, C.; Mikosch, T. Large deviations of heavy-tailed random sums with applications in insurance and finance. Journal of Applied Probability 34 (1997), no. 2, 293-308. Mikosch, T.; Nagaev, A. V. Large deviations of heavy-tailed sums with applications in insurance. Extremes 1 (1998), no. 1, 81-110. Nagaev, S. V. Letter to the editors: "Probabilistic inequalities for sums of independent random variables" (Theory of Probability and its Applications 16 (1971), 643-660) by D. H. Fuk and S. V. Nagaev. Theory of Probability and its Applications 21 (1976), no.4, 896. Ng, K. W.; Tang, Q.; Yan, J.; Yang, H. Precise large deviations for sums of random variables with consistently varying tails. Journal of Applied Probability 41 (2004), no.1, 93-107. Schlegel, S. Ruin probabilities in perturbed risk models. Insurance: Mathematics & Economics 22 (1998), no. 1, 93-104. Stein, C. A note on cumulative sums. Annals of Mathematical Statistics 17 (1946), no.4, 498-499. Tang, Q. Asymptotics for the finite time ruin probability in the renewal model with consistent variation. Stochastic Models 20 (2004), no. 3, 281-297. Tang, Q.; Su, C.; Jiang, T.; Zhang, J. Large deviations for heavy-tailed random sums in compound renewal model. Statistics & Probability Letters 52 (2001), no. 1, 91-100. Tang, Q.; Tsitsiashvili, G. Precise estimates for the ruin probability infinite horizon in adiscrete-time model with heavy-tailed insurance and financial risks. Stochastic Processes and their Applications 108 (2003), no. 2, 299-325. |

Repository Staff Only: item control page