IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i1p217-232.html
   My bibliography  Save this article

Maximums on trees

Author

Listed:
  • Jelenković, Predrag R.
  • Olvera-Cravioto, Mariana

Abstract

We study the minimal/endogenous solution R to the maximum recursion on weighted branching trees given by R=D(⋁i=1NCiRi)∨Q, where (Q,N,C1,C2,…) is a random vector with N∈N∪{∞}, P(|Q|>0)>0 and nonnegative weights {Ci}, and {Ri}i∈N is a sequence of i.i.d. copies of R independent of (Q,N,C1,C2,…); =D denotes equality in distribution. Furthermore, when Q>0 this recursion can be transformed into its additive equivalent, which corresponds to the maximum of a branching random walk and is also known as a high-order Lindley equation. We show that, under natural conditions, the asymptotic behavior of R is power-law, i.e., P(|R|>x)∼Hx−α, for some α>0 and H>0. This has direct implications for the tail behavior of other well known branching recursions.

Suggested Citation

  • Jelenković, Predrag R. & Olvera-Cravioto, Mariana, 2015. "Maximums on trees," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 217-232.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:217-232
    DOI: 10.1016/j.spa.2014.09.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414914002099
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2014.09.004?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. Buraczewski, Dariusz & Damek, Ewa & Mentemeier, Sebastian & Mirek, Mariusz, 2013. "Heavy tailed solutions of multivariate smoothing transforms," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1947-1986.
    2. Biggins, J. D., 1998. "Lindley-type equations in the branching random walk," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 105-133, June.
    3. Karpelevich, F. I. & Kelbert, M. Ya. & Suhov, Yu. M., 1994. "Higher-order Lindley equations," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 65-96, September.
    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. Mariana Olvera-Cravioto & Octavio Ruiz-Lacedelli, 2021. "Stationary Waiting Time in Parallel Queues with Synchronization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 1-27, February.

    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. Fraiman, Nicolas & Lin, Tzu-Chi & Olvera-Cravioto, Mariana, 2023. "Stochastic recursions on directed random graphs," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    2. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.
    3. Mariana Olvera-Cravioto & Octavio Ruiz-Lacedelli, 2021. "Stationary Waiting Time in Parallel Queues with Synchronization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 1-27, February.
    4. Onno Boxma & Andreas Löpker & Michel Mandjes & Zbigniew Palmowski, 2021. "A multiplicative version of the Lindley recursion," Queueing Systems: Theory and Applications, Springer, vol. 98(3), pages 225-245, August.
    5. Iksanov, Alexander & Kolesko, Konrad & Meiners, Matthias, 2019. "Stable-like fluctuations of Biggins’ martingales," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4480-4499.
    6. Bassetti, Federico & Matthes, Daniel, 2014. "Multi-dimensional smoothing transformations: Existence, regularity and stability of fixed points," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 154-198.
    7. Biggins, J. D., 1998. "Lindley-type equations in the branching random walk," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 105-133, June.

    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:125:y:2015:i:1:p:217-232. 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.