From additive to second-order processes

The familiar Poisson process is a member of a class of stochastic processes known as additive processes. This broad class also contains the birth-death processes. Second-order processes are processes with two moments finite. The class of second-order processes includes the well-known weakly stationary as well as harmonizable processes. A natural evolution of concepts linking the class of additive processes and the class of second-order processes will be detailed. The connection arises via stable processes and random measures