IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v127y2017i1p80-106.html
   My bibliography  Save this article

Fractionally integrated inverse stable subordinators

Author

Listed:
  • Iksanov, Alexander
  • Kabluchko, Zakhar
  • Marynych, Alexander
  • Shevchenko, Georgiy

Abstract

A fractionally integrated inverse stable subordinator (FIISS) is the convolution of an inverse stable subordinator, also known as a Mittag-Leffler process, and a power function. We show that the FIISS is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean ‘inter-shot’ distribution and regularly varying response function. We prove local Hölder continuity of FIISS and a law of iterated logarithm for both small and large times.

Suggested Citation

  • Iksanov, Alexander & Kabluchko, Zakhar & Marynych, Alexander & Shevchenko, Georgiy, 2017. "Fractionally integrated inverse stable subordinators," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 80-106.
  • Handle: RePEc:eee:spapps:v:127:y:2017:i:1:p:80-106
    DOI: 10.1016/j.spa.2016.06.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414916300783
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spa.2016.06.001?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. Iksanov, Alexander & Kabluchko, Zakhar & Marynych, Alexander, 2016. "Weak convergence of renewal shot noise processes in the case of slowly varying normalization," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 67-77.
    2. Iksanov, Alexander, 2013. "Functional limit theorems for renewal shot noise processes with increasing response functions," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1987-2010.
    3. Nane, Erkan, 2009. "Laws of the iterated logarithm for a class of iterated processes," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1744-1751, August.
    4. Scalas, Enrico & Viles, Noèlia, 2014. "A functional limit theorem for stochastic integrals driven by a time-changed symmetric α-stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 385-410.
    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. Beghin, Luisa, 2018. "Fractional diffusion-type equations with exponential and logarithmic differential operators," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2427-2447.
    2. Iksanov, Alexander & Kabluchko, Zakhar & Marynych, Alexander, 2016. "Weak convergence of renewal shot noise processes in the case of slowly varying normalization," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 67-77.
    3. Yiqing Chen, 2019. "A Renewal Shot Noise Process with Subexponential Shot Marks," Risks, MDPI, vol. 7(2), pages 1-8, June.
    4. Endre Csáki & Antónia Földes, 2020. "Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2233-2257, December.
    5. Alsmeyer, Gerold & Iksanov, Alexander & Marynych, Alexander, 2017. "Functional limit theorems for the number of occupied boxes in the Bernoulli sieve," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 995-1017.
    6. Kobayashi, Kei, 2016. "Small ball probabilities for a class of time-changed self-similar processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 155-161.
    7. Gajda, Janusz & Magdziarz, Marcin, 2014. "Large deviations for subordinated Brownian motion and applications," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 149-156.
    8. Pang, Guodong & Zhou, Yuhang, 2018. "Functional limit theorems for a new class of non-stationary shot noise processes," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 505-544.
    9. Andreas S{o}jmark & Fabrice Wunderlich, 2023. "Functional CLTs for subordinated L\'evy models in physics, finance, and econometrics," Papers 2312.15119, arXiv.org, revised Jan 2024.
    10. Li, Bo & Pang, Guodong, 2022. "Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 285-339.
    11. Leonenko, N.N. & Papić, I. & Sikorskii, A. & Šuvak, N., 2017. "Heavy-tailed fractional Pearson diffusions," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3512-3535.

    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:eee:spapps:v:127:y:2017:i:1:p:80-106. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.