Expansion of the Distribution Functions Defining some Processes Based on the Poisson Distribution

The compound Poisson process in the wide sense is defined as a process for which the probability distribution of the number i of changes in the random function attached to the process, while the parameter passes from o to a fixed value x of the parameter measured on a suitable scale, is given by the Laplace-Stieltjes integral (variable) where U(v, x) for a fixed value of x defines the distribution of v. U(v, T) is called the risk distribution and is either T-independent or, dependent on x.