Το πρόβλημα της αναγωγής των μεικτών ανανεωτικών διαδικασιών σε μεικτές διαδικασίες Poisson και εφαρμογές
The problem of the reduction of mixed renewal processes to mixed Poisson process and applications
In this paper initially a characterization of Markov processes is presented as mixed renewal processes. The class of mixed renewal (MRPs for short) with a random vector mixing parameter, as defined by Lyberopoulos and Macheras (enlarging the original Huang’s class), is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. It is proven under a mild assumption that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as a mixing parameter, has a solution to a positive. This implies the equivalence of Markov processes, mixed Poisson processes and processes with the multinomial property within this class. In concrete examples, Markov’s property is identified by means of the above results. Another consequence is the invariance of the Markov property under certain changes of measures. A second important implementation of the above result is the equivalence of mixed Poisson processes with mixing parameter of a real-valued random with mixing probability distribution as well as to the Poisson process in the sense Huang mixed processes. In addition, there are some examples of ”canonical” probability spaces admiting counting processes, such that the equivalence of the above definitions is true. Finally, a Poisson mixed process characterization is given through regular conditional probability and it is shown that the assumptions of this characterization can not be omitted.