The Monotone Cumulants

In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in commutative, free and Boolean cases. The way we define (generalized) cumulants is so elementary that we need neither partition lattices nor generating functions. This new approach open the way to introduce “monotone cumulants” and we obtain quite simple proof of central limit theorem and Poisson’s law of small numbers in monotone probability theory.