  We present analogues of the Poisson limit distribution for the noncommutative
bm-independence, which is associated with several positive symmetric cones. We
construct related discrete Fock spaces with creation, annihilation and
conservation operators, and prove Poisson type limit theorems for them.
Properties of the positive cones, in particular the volume characteristic
property they enjoy, and the combinatorics of labelled noncrossing partitions,
play crucial role in these considerations.
