IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v20y2018i3d10.1007_s11009-016-9503-x.html
   My bibliography  Save this article

Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks

Author

Listed:
  • Tiandong Wang

    (Cornell University)

  • Sidney I. Resnick

    (Cornell University)

Abstract

Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity conditions. We extend these arguments to discrete mass functions and their associated measures using the concept that the mass function can be embedded in a joint density function with continuous arguments. We give two different conditions, monotonicity and convergence on the unit sphere, both of which can make the discrete function embeddable. Our results are then applied to the preferential attachment network model, and we conclude that the joint mass function of in- and out-degree is embeddable and thus regularly varying.

Suggested Citation

  • Tiandong Wang & Sidney I. Resnick, 2018. "Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 1029-1042, September.
    • Handle: RePEc:spr:metcap:v:20:y:2018:i:3:d:10.1007_s11009-016-9503-x
    DOI: 10.1007/s11009-016-9503-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-016-9503-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-016-9503-x?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. de Haan, L. & Omey, E., 1984. "Integrals and derivatives of regularly varying functions in d and domains of attraction of stable distributions II," Stochastic Processes and their Applications, Elsevier, vol. 16(2), pages 157-170, February.
    2. de Haan, L. & Omey, E. & Resnick, S., 1984. "Domains of attraction and regular variation in IRd," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 17-33, February.
    3. de Haan, L. & Resnick, S., 1987. "On regular variation of probability densities," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 83-93.
    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. Tiandong Wang & Panpan Zhang, 2022. "Directed hybrid random networks mixing preferential attachment with uniform attachment mechanisms," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(5), pages 957-986, October.

    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. Yun, Seokhoon, 1997. "On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity," Journal of Multivariate Analysis, Elsevier, vol. 63(2), pages 277-295, November.
    2. Li, Haijun & Hua, Lei, 2015. "Higher order tail densities of copulas and hidden regular variation," Journal of Multivariate Analysis, Elsevier, vol. 138(C), pages 143-155.
    3. Bikramjit Das & Tiandong Wang & Gengling Dai, 2022. "Asymptotic Behavior of Common Connections in Sparse Random Networks," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2071-2092, September.
    4. Peng, Liang, 2002. "Asymptotic expansions of densities of sums of random vectors without third moment," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 167-174, June.
    5. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    6. Omey, Edward & Vesilo, R., 2009. "Random Sums of Random Variables and Vectors," Working Papers 2009/09, Hogeschool-Universiteit Brussel, Faculteit Economie en Management.
    7. Mallor, F. & Omey, E. & Santos, J., 2007. "Multivariate weighted renewal functions," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 30-39, January.
    8. Cai, J. & Einmahl, J.H.J. & de Haan, L.F.M., 2011. "Estimation of extreme risk regions under multivariate regular variation," Other publications TiSEM b7a72a8d-f9bc-4129-ae9b-a, Tilburg University, School of Economics and Management.
    9. Yi He & John H. J. Einmahl, 2017. "Estimation of extreme depth-based quantile regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 449-461, March.
    10. Das, Bikramjit & Fasen-Hartmann, Vicky, 2024. "On heavy-tailed risks under Gaussian copula: The effects of marginal transformation," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    11. Vysotsky, Vladislav, 2010. "On the probability that integrated random walks stay positive," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1178-1193, July.
    12. Omey, Edward & Van Gulck, Stefan, 2015. "Intuitive approximations in discrete renewal theory, Part 1: Regularly varying case," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 68-74.
    13. Cai, J., 2012. "Estimation concerning risk under extreme value conditions," Other publications TiSEM a92b089f-bc4c-41c2-b297-c, Tilburg University, School of Economics and Management.
    14. Anand Deo & Karthyek Murthy, 2020. "Optimizing tail risks using an importance sampling based extrapolation for heavy-tailed objectives," Papers 2008.09818, arXiv.org.

    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:metcap:v:20:y:2018:i:3:d:10.1007_s11009-016-9503-x. 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.