IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v123y2013i6p1947-1986.html
   My bibliography  Save this article

Heavy tailed solutions of multivariate smoothing transforms

Author

Listed:
  • Buraczewski, Dariusz
  • Damek, Ewa
  • Mentemeier, Sebastian
  • Mirek, Mariusz

Abstract

Let N>1 be a fixed integer and (C1,…,CN,Q) a random element of M(d×d,R)N×Rd. We consider solutions of multivariate smoothing transforms, i.e. random variables R satisfying R=d∑i=1NCiRi+Q where =d denotes equality in distribution, and R,R1,…,RN are independent identically distributed Rd-valued random variables, and independent of (C1,…,CN,Q). We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our results also cover the case of complex valued weights (C1,…,CN).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:6:p:1947-1986
    DOI: 10.1016/j.spa.2013.02.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414913000410
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2013.02.003?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. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
    2. Caliebe, Amke & Rösler, Uwe, 2003. "Fixed points with finite variance of a smoothing transformation," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 105-129, 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. Jelenković, Predrag R. & Olvera-Cravioto, Mariana, 2015. "Maximums on trees," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 217-232.
    2. 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.

    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, Aleksander M., 2004. "Elementary fixed points of the BRW smoothing transforms with infinite number of summands," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 27-50, November.
    2. Olvera-Cravioto, Mariana, 2012. "Tail behavior of solutions of linear recursions on trees," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1777-1807.
    3. Decrouez, Geoffrey & Hambly, Ben & Jones, Owen Dafydd, 2015. "The Hausdorff spectrum of a class of multifractal processes," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1541-1568.
    4. Buraczewski, D. & Damek, E. & Zienkiewicz, J., 2018. "Pointwise estimates for first passage times of perpetuity sequences," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2923-2951.
    5. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    6. 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.
    7. 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.
    8. Ghorbel, M. & Huillet, T., 2007. "Additional aspects of the non-conservative Kolmogorov–Filippov fragmentation model," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1569-1583.
    9. Bassetti, Federico & Ladelli, Lucia, 2023. "Central limit theorem in uniform metrics for generalized Kac equations," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    10. Bertoin, Jean, 2008. "Asymptotic regimes for the occupancy scheme of multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1586-1605, September.
    11. Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    12. Yang, Hairuo, 2023. "On the law of terminal value of additive martingales in a remarkable branching stable process," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 361-376.
    13. Buraczewski, Dariusz, 2009. "On tails of fixed points of the smoothing transform in the boundary case," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3955-3961, November.
    14. Quansheng Liu & Emmanuel Rio & Alain Rouault, 2003. "Limit Theorems for Multiplicative Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 971-1014, October.
    15. Liang, Xingang & Liu, Quansheng, 2020. "Regular variation of fixed points of the smoothing transform," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4104-4140.
    16. Najmeddine Attia, 2014. "On the Multifractal Analysis of the Branching Random Walk in $$\mathbb{R }^d$$ R d," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1329-1349, December.
    17. Yueyun Hu, 2017. "Local Times of Subdiffusive Biased Walks on Trees," Journal of Theoretical Probability, Springer, vol. 30(2), pages 529-550, June.
    18. Bertoin, Jean, 2006. "Different aspects of a random fragmentation model," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 345-369, March.
    19. Xiaoqiang Wang & Chunmao Huang, 2017. "Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 961-995, September.
    20. Liu, Quansheng & Watbled, Frédérique, 2009. "Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3101-3132, October.

    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:123:y:2013:i:6:p:1947-1986. 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.