IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v35y2022i3d10.1007_s10959-021-01102-8.html
   My bibliography  Save this article

Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments

Author

Listed:
  • Christian Mönch

    (Institut für Mathematik Johannes Gutenberg-Universität)

Abstract

We show that $$\mathbb {P}( \ell _X(0,T] \le 1)=(c_X+o(1))T^{-(1-H)}$$ P ( ℓ X ( 0 , T ] ≤ 1 ) = ( c X + o ( 1 ) ) T - ( 1 - H ) , where $$\ell _X$$ ℓ X is the local time measure at 0 of any recurrent H-self-similar real-valued process X with stationary increments that admits a sufficiently regular local time and $$c_X$$ c X is some constant depending only on X. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound $$1-H$$ 1 - H on the decay exponent of $$\mathbb {P}( \ell _X(0,T] \le 1)$$ P ( ℓ X ( 0 , T ] ≤ 1 ) . Our approach establishes a new connection between persistence probabilities and Palm theory for self-similar random measures, thereby providing a general framework which extends far beyond the Gaussian case.

Suggested Citation

  • Christian Mönch, 2022. "Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1842-1862, September.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01102-8
    DOI: 10.1007/s10959-021-01102-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-021-01102-8
    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/s10959-021-01102-8?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. Thorisson, Hermann, 1995. "On time- and cycle-stationarity," Stochastic Processes and their Applications, Elsevier, vol. 55(2), pages 183-209, February.
    2. Taqqu, Murad S., 1978. "A representation for self-similar processes," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 55-64, March.
    3. Aurzada, Frank & Guillotin-Plantard, Nadine & Pène, Françoise, 2018. "Persistence probabilities for stationary increment processes," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1750-1771.
    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. Chi, Zhiyi, 2001. "Stationary self-similar random fields on the integer lattice," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 99-113, January.
    2. Nieuwenhuis, G., 1996. "Ergodicity Conditions and Cesaro Limit Results for Marked Point Processes," Research Memorandum 736, Tilburg University, School of Economics and Management.
    3. Nieuwenhuis, G., 1996. "Ergodicity Conditions and Cesaro Limit Results for Marked Point Processes," Other publications TiSEM e9963ce3-420e-49b5-88cc-9, Tilburg University, School of Economics and Management.
    4. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.
    5. Aurzada, Frank & Buck, Micha & Kilian, Martin, 2020. "Penalizing fractional Brownian motion for being negative," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6625-6637.
    6. Bardet, Jean-Marc & Tudor, Ciprian, 2014. "Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 1-16.
    7. Aurzada, Frank & Mukherjee, Sumit, 2023. "Persistence probabilities of weighted sums of stationary Gaussian sequences," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 286-319.
    8. Last, Günter, 1996. "Coupling with compensators," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 147-170, December.
    9. Bardhan, Indrajit, 1995. "Further applications of a general rate conservation law," Stochastic Processes and their Applications, Elsevier, vol. 60(1), pages 113-130, November.

    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:35:y:2022:i:3:d:10.1007_s10959-021-01102-8. 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.