Large deviations for heavy-tailed random sums in compound renewal model.

*(English)*Zbl 0977.60034The object of interest is a doubly stochastic sum \(S_t = \sum_{i=1}^{N(t)} X_i\), where \(N(t)\) is a random variable with finite expectation \(\lambda(t)\), and \(\lambda(t)\) goes to infinity for \(t \to \infty\), and the random variables \(X_i\) are of extended regular variation. The authors determine the tail behaviour of this sum, and relax the assumptions of a theorem of Klüppelberg and Mikosch who considered the same question. Applications to compound renewal models are given.

Reviewer: Nina Gantert (Berlin)

##### MSC:

60F10 | Large deviations |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

60F05 | Central limit and other weak theorems |

60G50 | Sums of independent random variables; random walks |

##### Keywords:

compound renewal risk model; extended regular variation; large deviations; renewal counting process
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\textit{Q. Tang} et al., Stat. Probab. Lett. 52, No. 1, 91--100 (2001; Zbl 0977.60034)

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##### References:

[1] | Cline, D.B.H., Hsing, T., 1991. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint, Texas A&M University. |

[2] | Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling extremal events for insurance and finance, (1997), Springer Berlin · Zbl 0873.62116 |

[3] | Fuc, D.H., Nagaev, S.V., 1971. Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16, 660-675 (in Russian). |

[4] | Heyde, C.C., A contribution to the theory of large deviations for sums of independent random variables, Z. wahrscheinlichkeitsth, 7, 303-308, (1967) · Zbl 0158.17001 |

[5] | Heyde, C.C., On large deviation problems for sums of random variables which are not attracted to the normal law, Ann. math. statist., 38, 1575-1578, (1967) · Zbl 0189.51704 |

[6] | Heyde, C.C., On large deviation probabilities in the case of attraction to a nonnormal stable law, Sankyá, 30, 253-258, (1968) · Zbl 0182.22903 |

[7] | Klüppelberg, C.; Mikosch, T., Large deviations of heavy-tailed random sums with applications in insurance and finance, J. appl. probab., 34, 293-308, (1997) · Zbl 0903.60021 |

[8] | Nagaev, A.V., 1969a. Integral limit theorems for large deviations when Cramer’s condition is not fulfilled I, II. Theory Probab. Appl. 14, 51-64, 193-208. · Zbl 0196.21002 |

[9] | Nagaev, A.V., 1969b. Limit theorems for large deviations when Cramer’s conditions are violated. Fiz-Mat. Nauk. 7, 17-22 (in Russian). |

[10] | Nagaev, S.V., 1973. Large deviations for sums of independent random variables. In: Transactions of the Sixth Prague Conference on Information Theory, Random Processes and Statistical Decision Functions. Academic, Prague, pp. 657-674. |

[11] | Nagaev, S.V., Large deviations of sums of independent random variables, Ann. probab., 7, 745-789, (1979) · Zbl 0418.60033 |

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