On the structure of a class of distributions obeying the principle of a single big jump

• Autorzy: Xu, H. , Scheutzow, M. , Wang, Y. , Cui, Z.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we present several heavy-tailed distributions belonging to the new class J of distributions obeying the principle of a single big jump introduced by Beck et al. (2015). We describe the structure of this class from different angles. First, we show that heavy-tailed distributions in the class J are automatically strongly heavy-tailed and thus have tails which are not too irregular. Second, we show that such distributions are not necessarily weakly tail equivalent to a subexponential distribution. We also show that the class of heavy-tailed distributions in J which are neither long-tailed nor dominatedly-varying-tailed is not only non-empty but even quite rich in the sense that it has a non-empty intersection with several other well-established classes. In addition, the integrated tail distribution of some particular of these distributions shows that the Pakes-Veraverbeke-Embrechts theorem for the class J does not hold trivially.

Słowa kluczowe

EN

principle of a single big jump; strongly heavy-tailed distribution; weak tail equivalence; integrated tail distribution

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2016
• Tom: Vol. 36, Fasc. 1
• Strony: 121--135
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Xu, H.

• School of Mathematical Sciences, Soochow University,

autor

Scheutzow, M.

• Institut für Mathematik, Technische Universität Berlin,

autor

Wang, Y.

• School of Mathematical Sciences, Soochow University,

autor

Cui, Z.

• School of Mathematics and Statistics, Changshu Institute of Technology,

Bibliografia

• [1] S. Beck, J. Blath, and M. Scheutzow, A new class of large claim size distributions: Definition, properties, and ruin theory, Bernoulli 21 (4) (2015), pp. 2457-2483.
• [2] P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, 1997.
• [3] S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, second edition, Springer, 2013.
• [4] C. Klüppelberg, On subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988), pp. 132-141.
• [5] C. Klüppelberg, Asymptotic ordering of distribution functions and convolution semigroups, Semigroup Forum 40 (1990), pp. 77-92.
• [6] T. Shimura and T. Watanabe, Infinite divisibility and generalized subexponentiality, Bernoulli 11 (2005), pp. 445-469.
• [7] Y. Wang, F. Cheng, and Y. Yang, Dominant relations on some subclasses of heavy-tailed distributions and their applications, Chinese J. Appl. Probab. Statist. 21 (1) (2005), pp. 21-30 (in Chinese).
• [8] H. Xu, M. Scheutzow, and Y. Wang, On a transformation between distributions obeying the principle of a single big jump, J. Math. Anal. Appl. 430 (2015), pp. 672-684.

Uwagi

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