In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined, and a moment-cumulant formula is derived in the bi-monotonic setting. In general, the bi-monotonic product of states is not a state and the bi-monotonic convolution of probability measures on the plane is not a probability measure. This provides an additional example of how positivity need not be preserved under conditional bi-free convolutions.
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T.H. is supported by JSPS Grant-in-Aid for Young Scientists (B) 15K17549 and (A) 17H04823. P.S. is supported by NSERC (Canada) Grant RGPIN-2017-05711. The authors are grateful to Malte Gerhold for pointing out an error in a previous version of manuscript.
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Gu, Y., Hasebe, T. & Skoufranis, P. Bi-monotonic Independence for Pairs of Algebras. J Theor Probab 33, 533–566 (2020). https://doi.org/10.1007/s10959-019-00884-2
- Bi-monotonic independence
- Bi-monotonic cumulants
- Bi-monotonic convolution
Mathematics Subject Classification (2010)
- Primary 46L53
- Secondary 46L54