A bivariate Lévy process with negative binomial and gamma marginals

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 Lévy 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.