On a transformation of the weighted compound Poisson process

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Abstract
We are using the following terminology—essentially following Feller:a) Compound Poisson VariableThis is a random variable where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variablehence (common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functions b) Weighted Compound Poisson VariableThis is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ) hence or in the language of characteristic functions Let [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads then It is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.

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