Some properties of compound Poisson-Geometric distribution

In the classical risk models,the homogeneous Poisson process is used to describe the number of claims.However,the risk event and claim event are not equivalent in the practical insurance,Poisson-Geometric distribution is more appropriate for describing the claim numbers than Poisson distribution.Using the technique of probability generating function,we study the closure property of compound Poisson-Geometric distributions under convolution.We discuss the relationship between the compound Poisson-Geometric distribution and compound Poisson distribution.The relationship between compound generalized negative binomial distribution and compound Poisson-Geometric distribution is also investigated.