Stability equations for processes with stationary independent increments using branching processes and Poisson mixtures

The equation X1X2W( X1+ X2)with W uniform (0,1) distributed and W,X1 and X2 independent, is generalized in several directions. Most importantly, a generalized multiplication operation is used in which subcritical branching processes, both with discrete and continuous state space, play an important role. The solutions of the equations so obtained are related to the concepts of self-decomposability and stability, both in the classical and in an extended sense. The solutions for +-valued random variables are obtained from those for +-valued random variables by way of Poisson mixtures. There are also some new results on (generalized) unimodality.