  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 the commutative, free and Boolean cases. The way we define
(generalized) cumulants needs neither partition lattices nor generating
functions and then will give a new viewpoint to cumulants. We define ``monotone
cumulants'' in the sense of generalized cumulants and we obtain quite simple
proofs of central limit theorem and Poisson's law of small numbers in monotone
probability theory. Moreover, we clarify a combinatorial structure of
moment-cumulant formula with the use of ``monotone partitions''.
