Conditionally monotone independence I: Independence, additive convolutions and related convolutions

Details Conditionally monotone independence I: Independence, additive convolutions and related convolutions Conditionally monotone independence I: Independence, additive convolutions and related convolutions

2009-07-31

arxiv.org/abs/0907.5473v4

Mathematics

Operator Algebras

41 pages; small mistakes revised; to appear in Infin. Dimens. Anal. Quantum Probab. Relat. Top

Scientific paper

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.

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