IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v30y2017i4d10.1007_s10959-016-0696-2.html
   My bibliography  Save this article

Asymptotic Expansion for the Distribution Density Function of the Compound Poisson Process in Large Deviations

Author

Listed:
  • Aurelija Kasparavičiūtė

    (Vilnius Gediminas Technical University)

  • Dovilė Deltuvienė

    (Vilnius Gediminas Technical University)

Abstract

The paper is devoted to obtaining the asymptotic expansion and determination of the structure of the remainder term taking into consideration large deviations in the Cramér zone for the distribution density function of the standardized compound Poisson process. Following Deltuvienė and Saulis (Acta Appl Math 78:87–97, 2003. doi: 10.1023/A:1025783905023 ; Lith Math J 41:620–625, 2001) and Saulis and Statulevičius [Limit theorems for large deviations. Mathematics and its applications (Soviet Series), vol 73, pp 154–187, Kluwer, Dordrecht, 1991], the solution to the problem is achieved by first using a general lemma presented by Saulis (see Lemma 6.1 in Saulis and Statulevičius 1991, p. 154) on the asymptotic expansion for the density function of an arbitrary random variable with zero mean and unit variance and combining methods for cumulants and characteristic functions. By taking into consideration the large deviations in the Cramér zone for the density function of the standardized compound Poisson process, the result for the asymptotic expansion extends the asymptotic expansions for the density function of the sums of non-random number of summands (Deltuvienė and Saulis 2003, 2001).

Suggested Citation

  • Aurelija Kasparavičiūtė & Dovilė Deltuvienė, 2017. "Asymptotic Expansion for the Distribution Density Function of the Compound Poisson Process in Large Deviations," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1655-1676, December.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:4:d:10.1007_s10959-016-0696-2
    DOI: 10.1007/s10959-016-0696-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-016-0696-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-016-0696-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Gutti Babu & Kesar Singh & Yaning Yang, 2003. "Edgeworth expansions for compound Poisson processes and the bootstrap," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(1), pages 83-94, March.
    2. Hanna Döring & Peter Eichelsbacher, 2013. "Moderate Deviations via Cumulants," Journal of Theoretical Probability, Springer, vol. 26(2), pages 360-385, June.
    Full references (including those not matched with items on IDEAS)

    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. Hafouta, Yeor, 2023. "Convergence rates in the functional CLT for α-mixing triangular arrays," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 242-290.
    2. O. V. Chernoyarov & A. S. Dabye & F. N. Diop & Y. A. Kutoyants, 2022. "Non asymptotic expansions of the MME in the case of Poisson observations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(8), pages 927-950, November.
    3. Deng Zhang, 2017. "Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1076-1103, September.
    4. Marek Skarupski & Qinhao Wu, 2024. "Confidence bounds for compound Poisson process," Statistical Papers, Springer, vol. 65(8), pages 5351-5377, October.
    5. Deng Zhang, 2019. "Gaussian Fluctuations and Moderate Deviations of Eigenvalues in Unitary Invariant Ensembles," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1647-1687, December.
    6. Bing-Yi Jing & Qiying Wang & Wang Zhou, 2015. "Cramér-Type Moderate Deviation for Studentized Compound Poisson Sum," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1556-1570, December.
    7. Peter Eichelsbacher & Matthias Löwe, 2019. "Lindeberg’s Method for Moderate Deviations and Random Summation," Journal of Theoretical Probability, Springer, vol. 32(2), pages 872-897, 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:spr:jotpro:v:30:y:2017:i:4:d:10.1007_s10959-016-0696-2. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.