  We define a product of algebraic probability spaces equipped with two states.
This product is called a conditionally monotone product. This product is a new
example of independence in non-commutative probability theory and unifies the
monotone and Boolean products, and moreover, the orthogonal product. Then we
define the associated cumulants and calculate the limit distributions in
central limit theorem and Poisson's law of small numbers. We also prove a
combinatorial moment-cumulant formula using monotone partitions. We investigate
some other topics such as infinite divisibility for the additive convolution
and deformations of the monotone convolution. We define cumulants for a general
convolution to analyze the deformed convolutions.
