Tail asymptotics of light-tailed Weibull-like sums

• Autorzy: Asmussen S. , Hashorva E. , Laub P. J. , Taimre T.

Identyfikatory

DOI

10.19195/0208-4147.37.2.3

• Języki publikacji: EN

Abstrakty

EN

We consider sums of n i.i.d. random variables with tails close to exp{−x^β} for some β > 1. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg, and Resnick (1993) are discussed from the point of view of tails rather than of densities, using a somewhat different angle, and supplemented with bounds, results on a random number N of terms, and simulation algorithms.

Słowa kluczowe

EN

Gumbel MDA; Laplace transform; rareevent simulation; sums of random variables; Weibull distribution

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2017
• Tom: Vol. 37, Fasc. 2
• Strony: 235--256
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Asmussen S.

• Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark

autor

Hashorva E.

• Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland

autor

Laub P. J.

• University of Queensland & Aarhus University, Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia

autor

Taimre T.

• Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia

Bibliografia

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• [6] A. A. Balkema and P. Embrechts, High Risk Scenarios and Extremes: A Geometric Approach, European Mathematical Society, Amsterdam 2007.
• [7] A. A. Balkema, C. Klüppelberg, and S. I. Resnick, Densities with Gaussian tails, Proc. Lond. Math. Soc. 66 (1993), pp. 568-588.
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• [11] P. Embrechts, E. Hashorva, and T. Mikosch, Aggregation of log-linear risks, J. Appl. Probab. 51A (2014), pp. 203-212.
• [12] P. Embrechts, C. Klüppelberg, and T. Mikosch, Modeling Extreme Events for Insurance and Finance, Springer, 1997.
• [13] E. Hashorva and J. Li, Asymptotics for a discrete-time risk model with the emphasis on financial risk, Probab. Engrg. Inform. Sci. 28 (2014), pp. 573-588.
• [14] J. L. Jensen, Saddlepoint Approximations, Clarendon Press, Oxford 1995.
• [15] S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, 2008.
• [16] H. Rootzén, A ratio limit theorem for the tails of weighted sums, Ann. Probab. 15 (1987), pp. 728-747.
• [17] T. Watanabe, Convolution equivalence and distributions of random sums, Probab. Theory Related Fields 142 (2008), pp. 367-397.

Uwagi

This paper is dedicated to Tomasz Rolski in appreciation of his fundamental contributions to applied probability over a lifetime, and of his sustaining friendship.

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).