On the Moment Characteristics for the Univariate Compound Poisson and Bivariate Compound Poisson Processes with Applications

The univariate and bivariate compound Poisson process (CPP and BCPP, respectively) ensure a better description than the homogeneous Poisson process for clustering of events. In this paper, new explicit representations of the moment characteristics (general, central, factorial, binomial and ordinary moments, factorial cumulants) and some covariance structures are derived for the CPP and BCPP. Then, the skewness and kurtosis of the univariate CPP are obtained for the first time and special cases of the CPP are studied in detail. Applications to two real data sets are given to illustrate the usage of these processes.