IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i4p2228-2249.html
   My bibliography  Save this article

Trimmed Lévy processes and their extremal components

Author

Listed:
  • Ipsen, Yuguang
  • Maller, Ross
  • Resnick, Sidney

Abstract

We analyze a stochastic process of the form (r)Xt=Xt−∑i=1rΔt(i), where (Xt)t≥0 is a driftless, infinite activity, subordinator on R+ with its jumps on [0,t] ordered as Δt(1)≥Δt(2)⋯. The r largest of these are “trimmed” from Xt to give (r)Xt. When r→∞, both (r)Xt↓0 and Δt(r)↓0 a.s. for each t>0, and it is interesting to study the weak limiting behavior of ((r)Xt,Δt(r)) in this case. We term this “large-trimming” behavior, and study the joint convergence of ((r)Xt,Δt(r)) as r→∞ under linear normalization, assuming extreme value-related conditions on the Lévy measure of Xt which guarantee that Δt(r) has a limit distribution with linear normalization. Allowing (r)Xt to have random centering and norming in a natural way, we first show that ((r)Xt,Δt(r)) has a bivariate normal limiting distribution as r→∞; then replacing the random normalizations with deterministic normings produces normal, and in some cases, non-normal, limits whose parameters we can specify.

Suggested Citation

  • Ipsen, Yuguang & Maller, Ross & Resnick, Sidney, 2020. "Trimmed Lévy processes and their extremal components," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2228-2249.
  • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2228-2249
    DOI: 10.1016/j.spa.2019.06.018
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414919300559
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spa.2019.06.018?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. Ipsen, Yuguang & Maller, Ross & Resnick, Sidney, 2019. "Ratios of ordered points of point processes with regularly varying intensity measures," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 205-222.
    2. De Haan, Laurens, 1974. "Equivalence classes of regularly varying functions," Stochastic Processes and their Applications, Elsevier, vol. 2(3), pages 243-259, July.
    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. 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. de Haan, L. & Resnick, S. I., 1979. "Local Limit Theorems for Sample Extremes," Econometric Institute Archives 272194, Erasmus University Rotterdam.
    2. Omey, Edward & Segers, Johan, 2009. "Generalised regular variation of arbitrary order," Working Papers 2009/02, Hogeschool-Universiteit Brussel, Faculteit Economie en Management.
    3. A. Berlinet & A. Elamine & A. Mas, 2011. "Local linear regression for functional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 1047-1075, October.
    4. Saulius Paukštys & Jonas Šiaulys & Remigijus Leipus, 2023. "Truncated Moments for Heavy-Tailed and Related Distribution Classes," Mathematics, MDPI, vol. 11(9), pages 1-15, May.

    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:130:y:2020:i:4:p:2228-2249. 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.