A bivariate Lévy process with negative binomial and gamma marginals

  • Authors:

  • Tomasz J. Kozubowski;Anna K. Panorska;Krzysztof Podgórski

  • Affiliations:

  • Department of Mathematics and Statistics, University of Nevada, Mail Stop 84, 89557-0045 Reno, NV, United States;Department of Mathematics and Statistics, University of Nevada, Mail Stop 84, 89557-0045 Reno, NV, United States;Lund University, Sweden

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2008

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Abstract

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.