Increasing C-Additive Processes

It is shown that any infinitely divisible distribution μ on R+ gives rise to a class of increasing additive processes we call C-additive processes, where C is a continuous semigroup of cumulant generating functions. The marginal and increment distributions of these pocesses are characterized in terms of their Lévy measure and their drift coefficient. Integral representations of C-additive processes in terms of a Poisson random measure are obtained. The limiting behavior (as t → ∞) of two subclasses of C-additive processes leads to new characterizations of C-selfdecomposable and C-stable distributions on R+.