A new class of multivariate counting processes and its characterization

In this paper, we suggest a new class of multivariate counting processes which generalizes and extends the multivariate generalized Polya process recently studied in Cha and Giorgio [On a class of multivariate counting processes, Adv. Appl. Probab. 48 (2016), pp. 443–462]. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We also discuss the dependence structure of the proposed multivariate counting process and other stochastic properties such as the joint distributions of the number of events in an arbitrary interval or disjoint intervals and the conditional joint distribution of the arrival times of different types of events given the number of events. The corresponding marginal processes are also characterized.