On some characterizations of the Poisson process

This paper gives some characterizations of Poisson processes, the model which it considers is inspired by Kimeldorf and Thall and the paper generalizes the results of Chandramohan and Liang. More precisely, it considers an arbitrarily delayed renewal process, at each arrival time the paper allows the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i=1,...,k. It will show that the existence of a pair of uncorrelated processes Di, Dj, iℝj, implies the renewal process is Poisson. Some other related characterization results are also obtained.