Discrete compound Poisson processes (namely nonnegative integer-valued Lévy processes) have the property that more than one event occurs in a small enough time interval. These stochastic processes produce the discrete compound Poisson distributions. In this article, we introduce ten approaches to prove the probability mass function of discrete compound Poisson distributions, and we obtain seven approaches to prove the probability mass function of Poisson distributions. Finally, we discuss the connection between additive functions in probabilistic number theory and discrete compound Poisson distributions and give a numerical example. Stuttering Poisson distributions (a special case of discrete compound Poisson distributions) are applied to numerical solution of optimal (s, S) inventory policies by using continuous approximation method.