A bivariate Lévy process with negative binomial and gamma marginals

The joint distribution of XX and NN, where NN has a geometric distribution and XX is the sum of NN IID exponential variables (independent of NN), is infinitely divisible. This leads to a bivariate Lévy process {(X(t),N(t)),t≥0}{(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))(X(t),N(t)) at a fixed time tt, 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.