A bivariate Levy process with negative binomial and gamma marginals

Research output: Contribution to journalArticle


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.


Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • 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
Original languageEnglish
Pages (from-to)1418-1437
JournalJournal of Multivariate Analysis
Issue number7
Publication statusPublished - 2008
Publication categoryResearch