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On a class of multivariate counting processes. (English) Zbl 1346.60066

Summary: In this paper, we define and study a new class of multivariate counting processes, named “multivariate generalized Pólya processes”. Initially, we define and study the bivariate generalized Pólya processes and briefly discuss their reliability applications. In order to derive the main properties of the process, we suggest some key properties and an important characterization of the processes. Due to these properties and the characterization, the main properties of the bivariate generalized Pólya processes are obtained efficiently. The marginal processes of the multivariate generalized Pólya processes are shown to be the univariate generalized Pólya processes studied in [the first author, Adv. Appl. Probab. 46, No. 4, 1148–1171 (2014; Zbl 1305.60088)]. Given the history of a marginal process, the conditional property of the other process is also discussed. The bivariate generalized Pólya processes are 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 processes.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62P30 Applications of statistics in engineering and industry; control charts

Citations:

Zbl 1305.60088
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