IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v127y2017i3p995-1017.html
   My bibliography  Save this article

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

Author

Listed:
  • Alsmeyer, Gerold
  • Iksanov, Alexander
  • Marynych, Alexander

Abstract

The Bernoulli sieve is the infinite Karlin “balls-in-boxes” scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals n, we prove several functional limit theorems (FLTs) in the Skorohod space D[0,1] endowed with the J1- or M1-topology for the number Kn∗(t) of boxes containing at most [nt] balls, t∈[0,1], and the random distribution function Kn∗(t)/Kn∗(1), as n→∞. The limit processes for Kn∗(t) are of the form (X(1)−X((1−t)−))t∈[0,1], where X is either a Brownian motion, a spectrally negative stable Lévy process, or an inverse stable subordinator. The small value probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for Kn∗(t)/Kn∗(1) is a Lévy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of Kn∗(1). First, for any Karlin occupancy scheme with deterministic probabilities (pk)k≥1, we obtain an approximation, uniformly in t∈[0,1], of the number of boxes with at most [nt] balls by a counting function defined in terms of (pk)k≥1. Second, we prove several FLTs for the number of visits to the interval [0,nt] by a perturbed random walk, as n→∞. If the stick-breaking factor has a beta distribution with parameters θ>0 and 1, the process (Kn∗(t))t∈[0,1] has the same distribution as a similar process defined by the number of cycles of length at most [nt] in a θ-biased random permutation a.k.a. a Ewens permutation with parameter θ. As a consequence, our FLT with Brownian limit forms a generalization of a FLT obtained earlier in the context of Ewens permutations by DeLaurentis and Pittel (1985), Hansen (1990), Donnelly et al. (1991), and Arratia and Tavaré (1992).

Suggested Citation

  • 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.
  • Handle: RePEc:eee:spapps:v:127:y:2017:i:3:p:995-1017
    DOI: 10.1016/j.spa.2016.07.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S030441491630120X
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spa.2016.07.007?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, 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.
    2. Iksanov, Alexander, 2012. "On the number of empty boxes in the Bernoulli sieve II," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2701-2729.
    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. Iksanov, Alexander & Jedidi, Wissem & Bouzeffour, Fethi, 2017. "A law of the iterated logarithm for the number of occupied boxes in the Bernoulli sieve," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 244-252.
    2. Durieu, Olivier & Samorodnitsky, Gennady & Wang, Yizao, 2020. "From infinite urn schemes to self-similar stable processes," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2471-2487.

    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. 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. Yiqing Chen, 2019. "A Renewal Shot Noise Process with Subexponential Shot Marks," Risks, MDPI, vol. 7(2), pages 1-8, June.
    3. 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.
    4. 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.

    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:3:p:995-1017. 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.