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

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  • 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
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    References listed on IDEAS

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    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. & Resnick, S., 1987. "On regular variation of probability densities," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 83-93.
    3. 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.
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    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.

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