On Poisson and Compound Poisson Processes and Some Comparisons

Abstract

Poisson and Compound Poisson processes are well known and have been studied and applied extensively. We define the random extrema process generated by the Poisson process and look at the distributional properties of the corresponding random variables. A few results are obtained by comparing Poisson and Compound Poisson processes with respect to stochastic orderings and based on the corresponding rate parameters and/ or the distribution function of summand random variables. We have presented detailed proofs of the results and their converse implications as applicable. We address these questions in this article: As the probability mass function of a Poisson random variable is log-concave, does this imply that the random maxima generated by a Poisson random variable also preserve this property? Whether more arrivals occurring for a Poisson process by a given time  when compared to another independent Poisson process? Some illustrations and applications are given along with the computations of the relevant quantities using the properties of the Poisson process.