Some convolution theorems for compound poisson processes

Abstract

Abstract Previously the term compound Poisson processes in the wide sense (cPp i. w. s.) has by the present author in several papers (e.g. 1964) been used for a process for which the probability of the occurrence of m changes in the interval (0, τ) on the absolute parameter scale, is defined by the following relation: where s and t are real-valued, positive functions of τ with one-to-one correspondence between t and τ, s and τ respectively. If the size of a change is a random variable, with the conditional distribution for the size of a change known to have occurred at t = u, V(x, u) with zu = z(η, u) as corresponding characteristic function, the distribution functions, defining the process, F(x; t, s), and their corresponding characteristic functions, ϕ(η; t, s) are defined by the following relations, where V (x, t) denotes the integral and, consequently, the characteristic function ξ corresponding to V (x, t), under mild regularity conditions equal to the expressions for V (x, t) and ξ may easily be extended to the case, where V(x, u) is a discontinuous function of u by substituting for V(x, u)du and zudu respectively. Further, in (2a) and in the following context, the asterisk power m* of any distribution function denotes for m = 0, unity, and, for m > 0 the m times iterated convolution of the function with itself.