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A bivariate Levy process with negative binomial and gamma marginals

Publiceringsår: 2008
Språk: Engelska
Sidor: 1418-1437
Publikation/Tidskrift/Serie: Journal of Multivariate Analysis
Volym: 99
Nummer: 7
Dokumenttyp: Artikel i tidskrift
Förlag: Academic Press


The joint distribution of X and N, where N has a geometric distribution and X is the sum of N IID exponential variables (independent of N), is infinitely divisible. This leads to a bivariate Levy process {(X(t), N(t)), t >= 0}, whose coordinates are correlated negative binomial and gamma processes. We derive basic properties of this process, including its covariance structure, representations, and stochastic self-similarity. We examine the joint distribution of (X(t), N(t)) at a fixed time t, along with the marginal and conditional distributions, joint integral transforms, moments, infinite divisibility, and stability with respect to random summation. We also discuss maximum likelihood estimation and simulation for this model.


  • Probability Theory and Statistics
  • operational time
  • random summation
  • random time transformation
  • stability
  • subordination self-similarity
  • negative binomial process
  • maximum likelihood estimation
  • divisibility
  • infinite
  • gamma Poisson process
  • discrete Levy process
  • gamma process


  • ISSN: 0047-259X

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