IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v15y2002i3d10.1023_a1016228101053.html
   My bibliography  Save this article

Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

Author

Listed:
  • R. A. Doney

    (University of Manchester)

  • R. A. Maller

    (University of Western Australia)

Abstract

We prove some limiting results for a Lévy process X t as t↓0 or t→∞, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t↓0; for example, we may have X t /t → P +∞ (t↓0) (weak divergence to +∞), whereas X t /t→∞ a.s. (t↓0) is impossible (both are possible when t→∞), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. “Almost sure stability” of X t , i.e., X t tending to a nonzero constant a.s. as t→∞ or as t↓0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to “strong law” behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.

Suggested Citation

  • R. A. Doney & R. A. Maller, 2002. "Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity," Journal of Theoretical Probability, Springer, vol. 15(3), pages 751-792, July.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:3:d:10.1023_a:1016228101053
    DOI: 10.1023/A:1016228101053
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1016228101053
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1016228101053?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. Harry Kesten & R. A. Maller, 1998. "Random Walks Crossing High Level Curved Boundaries," Journal of Theoretical Probability, Springer, vol. 11(4), pages 1019-1074, October.
    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. Søren Asmussen, 2022. "On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance," Finance and Stochastics, Springer, vol. 26(3), pages 383-416, July.
    2. Grahovac, Danijel, 2022. "Intermittency in the small-time behavior of Lévy processes," Statistics & Probability Letters, Elsevier, vol. 187(C).
    3. Yuguang Fan, 2017. "Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times," Journal of Theoretical Probability, Springer, vol. 30(2), pages 675-699, June.
    4. David M. Mason, 2021. "Self-Standardized Central Limit Theorems for Trimmed Lévy Processes," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2117-2144, December.

    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. R. A. Doney & R. A. Maller, 2007. "Almost Sure Relative Stability of the Overshoot of Power Law Boundaries," Journal of Theoretical Probability, Springer, vol. 20(1), pages 47-63, March.

    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:15:y:2002:i:3:d:10.1023_a:1016228101053. 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.