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Methodology and Computing in Applied Probability

, Volume 20, Issue 3, pp 1029–1042 | Cite as

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

  • Tiandong Wang
  • Sidney I. Resnick
Article
  • 43 Downloads

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.

Keywords

Multivariate regular variation Preferential attachment Random graphs Power laws In-degree Out-degree 

Mathematics Subject Classification (2010)

28A33 60G70 05C80 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

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