A bivariate Lévy process with negative binomial and gamma marginals

Tomas J. Kozubowski, Anna K. Panorska and Krzysztof Podgórski

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 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 study its bivariate marginal distributions of correlated gamma and negative binomial variables, and present their 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.

Key words:: 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