  We prove a multidimensional Poisson limit theorem in free probability, and
define joint free Poisson distributions in a non-commutative probability space.
We define (compound) free Poisson process explicitly, similar to the
definitions of (compound) Poisson processes in classical probability. We proved
that the sum of finitely many freely independent compound free Poisson
processes is a compound free Poisson processes. We give a step by step
procedure for constructing a (compound) free Poisson process. A Karhunen-Loeve
expansion theorem for centered free Poisson processes is proved. We generalize
free Poisson processes to a notion of free Poisson random measures (which is
slightly different from the previously defined ones in free probability, but
more like an analogue of classical Poisson random measures). Then we develop
the integration theory of real-valued functions with respect to a free Poisson
random measure, generalizing the classical integration theory to the free
probability case. We find that the integral of a function (in certain spaces of
functions) with respect to a free Poisson random measure has a compound free
Poisson distribution. For centered free Poisson random measures, we can get a
simpler and more beautiful integration theory.
