On a class of multivariate counting processes

In this paper we define and study a new class of multivariate counting processes, named 'multivariate generalized Pólya process'. Initially, we define and study the bivariate generalized Pólya process and briefly discuss its reliability application. In order to derive the main properties of the process, we suggest some key properties and an important characterization of the process. Due to these properties and the characterization, the main properties of the bivariate generalized Pólya process are obtained efficiently. The marginal processes of the multivariate generalized Pólya process are shown to be the univariate generalized Pólya processes studied in Cha (2014). Given the history of a marginal process, the conditional property of the other process is also discussed. The bivariate generalized Pólya process is extended to the multivariate case. We define a new dependence concept for multivariate point processes and, based on it, we analyze the dependence structure of the multivariate generalized Pólya process.