IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i21p4128-d964079.html
   My bibliography  Save this article

Local Closure under Infinitely Divisible Distribution Roots and Esscher Transform

Author

Listed:
  • Zhaolei Cui

    (School of Mathematics and Statistics, Changshu Institute of Technology, Suzhou 215000, China
    These authors contributed equally to this work.)

  • Yuebao Wang

    (School of Mathematical Sciences, Soochow University, Suzhou 215006, China
    These authors contributed equally to this work.)

  • Hui Xu

    (School of Mathematical Sciences, Soochow University, Suzhou 215006, China
    These authors contributed equally to this work.)

Abstract

In this paper, we show that the local distribution class L l o c ∩ OS l o c is not closed under infinitely divisible distribution roots, i.e., there is an infinitely divisible distribution which belongs to the class, while the corresponding Lévy distribution does not. Conversely, we give a condition, under which, if an infinitely divisible distribution belongs to the class L l o c ∩ OS l o c , then so does the Lévy distribution. Furthermore, we find some sufficient conditions that are more concise and intuitive. Using different methods, we also give a corresponding result for another local distribution class, which is larger than the above class. To prove the above results, we study the local closure under random convolution roots. In particular, we obtain a result on the local closure under the convolution root. In these studies, the Esscher transform of distribution plays a key role, which clarifies the relationship between these local distribution classes and related global distribution classes.

Suggested Citation

  • Zhaolei Cui & Yuebao Wang & Hui Xu, 2022. "Local Closure under Infinitely Divisible Distribution Roots and Esscher Transform," Mathematics, MDPI, vol. 10(21), pages 1-24, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4128-:d:964079
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/21/4128/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/21/4128/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Søren Asmussen & Serguei Foss & Dmitry Korshunov, 2003. "Asymptotics for Sums of Random Variables with Local Subexponential Behaviour," Journal of Theoretical Probability, Springer, vol. 16(2), pages 489-518, April.
    2. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    3. Jiang, Tao & Wang, Yuebao & Cui, Zhaolei & Chen, Yuxin, 2019. "On the almost decrease of a subexponential density," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 71-79.
    4. Wang, Yuebao & Yang, Yang & Wang, Kaiyong & Cheng, Dongya, 2007. "Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 256-266, March.
    5. Veraverbeke, Noel, 1993. "Asymptotic estimates for the probability of ruin in a Poisson model with diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 13(1), pages 57-62, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Saulius Paukštys & Jonas Šiaulys & Remigijus Leipus, 2023. "Truncated Moments for Heavy-Tailed and Related Distribution Classes," Mathematics, MDPI, vol. 11(9), pages 1-15, May.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jianxi Lin, 2012. "Second order Subexponential Distributions with Finite Mean and Their Applications to Subordinated Distributions," Journal of Theoretical Probability, Springer, vol. 25(3), pages 834-853, September.
    2. Toshiro Watanabe, 2022. "Embrechts–Goldie’s Problem on the Class of Lattice Convolution Equivalent Distributions," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2622-2642, December.
    3. Toshiro Watanabe & Kouji Yamamuro, 2010. "Local Subexponentiality and Self-decomposability," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1039-1067, December.
    4. Toshiro Watanabe & Kouji Yamamuro, 2017. "Two Non-closure Properties on the Class of Subexponential Densities," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1059-1075, September.
    5. Yu, Changjun & Wang, Yuebao & Yang, Yang, 2010. "The closure of the convolution equivalent distribution class under convolution roots with applications to random sums," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 462-472, March.
    6. Schlegel, Sabine, 1998. "Ruin probabilities in perturbed risk models," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 93-104, May.
    7. Yuebao Wang & Kaiyong Wang, 2009. "Equivalent Conditions of Asymptotics for the Density of the Supremum of a Random Walk in the Intermediate Case," Journal of Theoretical Probability, Springer, vol. 22(2), pages 281-293, June.
    8. Yu, Changjun & Wang, Yuebao & Cui, Zhaolei, 2010. "Lower limits and upper limits for tails of random sums supported on," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1111-1120, July.
    9. Geluk, J.L. & De Vries, C.G., 2006. "Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 39-56, February.
    10. Neveka M. Olmos & Emilio Gómez-Déniz & Osvaldo Venegas, 2022. "The Heavy-Tailed Gleser Model: Properties, Estimation, and Applications," Mathematics, MDPI, vol. 10(23), pages 1-16, December.
    11. Yang, Xiangfeng, 2015. "Exact upper tail probabilities of random series," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 13-19.
    12. Braverman, Michael, 1997. "Suprema and sojourn times of Lévy processes with exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 265-283, June.
    13. Wang, Yuebao & Yang, Yang & Wang, Kaiyong & Cheng, Dongya, 2007. "Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 256-266, March.
    14. Vaios Dermitzakis & Konstadinos Politis, 2011. "Asymptotics for the Moments of the Time to Ruin for the Compound Poisson Model Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 749-761, December.
    15. Shimizu, Yasutaka, 2009. "A new aspect of a risk process and its statistical inference," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 70-77, February.
    16. M. S. Sgibnev, 1998. "On the Asymptotic Behavior of the Harmonic Renewal Measure," Journal of Theoretical Probability, Springer, vol. 11(2), pages 371-382, April.
    17. Griffin, Philip S. & Maller, Ross A. & Schaik, Kees van, 2012. "Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 382-392.
    18. Sgibnev, M. S., 2001. "Exact asymptotic behaviour of the distribution of the supremum," Statistics & Probability Letters, Elsevier, vol. 52(3), pages 301-311, April.
    19. Chiu, S. N. & Yin, C. C., 2003. "The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 59-66, August.
    20. Yang Yang & Xinzhi Wang & Shaoying Chen, 2022. "Second Order Asymptotics for Infinite-Time Ruin Probability in a Compound Renewal Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1221-1236, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4128-:d:964079. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.