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
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Abstract
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.