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arXiv · 1009.1505

A three-state independence in non-commutative probability

Takahiro Hasebe
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Abstract

We define a new independence in non-commutative probability, called $\alpha$-freeness, with respect to a triplet of states. This concept unifies several independences in non-commutative probability, in particular, free, monotone, antimonotone and Boolean ones as well as conditionally free, conditionally monotone and conditionally antimonotone independences. Moreover, the associative law of $\alpha$-freeness is transferred to the other independences. As a consequence, $\alpha$-free cumulants unify the cumulants for free, monotone, antimonotone and Boolean independences. The central limit theorem for $\alpha$-freeness is computed. The limit distribution turns out to be a triplet of the Kesten distributions.

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Takahiro Hasebe. 2010-09-08. A three-state independence in non-commutative probability. https://arxiv.org/abs/1009.1505

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