CLASSES OF MULTIVARIATE EXPONENTIAL AND MULTIAVARIATE GEOMETRIC DISTRIBUTIONS DERIVED FROM MARKOV PROCESSES

Abstract

We define a class of multivariate exponential distributions as the distributions of occupancy times in upwards skip‐free Markov processes in continuous time. These distributions are infinitely divisible, and the multivariate gamma class defined by convolutions and fractions is a substantial generalization of the class defined by Johnson & Kotz (1972). Parallel classes of multivariate geometric and multivariate negative binomial distributions are constructed from occupancy times in ‘instant’ upwards skip‐free Markov chains. Maximum likelihood estimation and times series applications are discussed.