IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v108y2003i2p327-343.html
   My bibliography  Save this article

Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative

Author

Listed:
  • Hambly, B. M.
  • Kersting, G.
  • Kyprianou, A. E.

Abstract

We show that under a 3+[delta] moment condition (where [delta]>0) there exists a 'Hartman-Winter' Law of the iterated logarithm for random walks conditioned to stay non-negative. We also show that under a second moment assumption the conditioned random walk eventually grows faster than n1/2(log n)-(1+[epsilon]) for any [epsilon]>0 and yet slower than n1/2(log n)-1. The results are proved using three key facts about conditioned random walks. The first is the relation of its step distribution to that of the original random walk given by Bertoin and Doney (Ann. Probab. 22 (1994) 2152). The second is the pathwise construction in terms of excursions in Tanaka (Tokyo J. Math. 12 (1989) 159) and the third is a new Skorohod-type embedding of the conditioned process in a Bessel-3 process.

Suggested Citation

  • Hambly, B. M. & Kersting, G. & Kyprianou, A. E., 2003. "Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 327-343, December.
  • Handle: RePEc:eee:spapps:v:108:y:2003:i:2:p:327-343
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(03)00103-0
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Bertoin, Jean, 1993. "Splitting at the infimum and excursions in half-lines for random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 17-35, August.
    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. Cox, A. M. G. & Hobson, D. G., 2004. "An optimal Skorokhod embedding for diffusions," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 17-39, 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. Chaumont, L., 1996. "Conditionings and path decompositions for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 39-54, November.
    2. Ceren Vardar-Acar & Mine Çağlar & Florin Avram, 2021. "Maximum Drawdown and Drawdown Duration of Spectrally Negative Lévy Processes Decomposed at Extremes," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1486-1505, September.
    3. Fitzsimmons, P. J. & Getoor, R. K., 1995. "Occupation time distributions for Lévy bridges and excursions," Stochastic Processes and their Applications, Elsevier, vol. 58(1), pages 73-89, July.
    4. Pierre Andreoletti & Roland Diel, 2011. "Limit Law of the Local Time for Brox’s Diffusion," Journal of Theoretical Probability, Springer, vol. 24(3), pages 634-656, September.
    5. Shi, Zhan, 1998. "A local time curiosity in random environment," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 231-250, August.
    6. Ivanovs, Jevgenijs & Thøstesen, Jakob D., 2021. "Discretization of the Lamperti representation of a positive self-similar Markov process," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 200-221.
    7. Ivanovs, Jevgenijs, 2017. "Splitting and time reversal for Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2699-2724.
    8. Comets, Francis & Loukianov, Oleg & Loukianova, Dasha, 2024. "Limit of the environment viewed from Sinaï’s walk," Stochastic Processes and their Applications, Elsevier, vol. 168(C).
    9. Chaumont, Loïc & Rivero, Víctor, 2007. "On some transformations between positive self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1889-1909, December.

    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:108:y:2003:i:2:p:327-343. 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.