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.60E05 60E07 60F05 60G18 60G50 60G51 62H05 62H12 Discrete Lévy process Gamma process Gamma Poisson process Infinite divisibility Maximum likelihood estimation Negative binomial process Operational time Random summation Random time transformation Stability Subordination Self-similarity