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.
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