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2011 Joint cumulants for natural independence
Takahiro Hasebe, Hayato Saigo
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Electron. Commun. Probab. 16: 491-506 (2011). DOI: 10.1214/ECP.v16-1647


Many kinds of independence have been defined in non-commutative probability theory. Natural independence is an important class of independence; this class consists of five independences (tensor, free, Boolean, monotone and anti-monotone ones). In the present paper, a unified treatment of joint cumulants is introduced for natural independence. The way we define joint cumulants enables us not only to find the monotone joint cumulants but also to give a new characterization of joint cumulants for other kinds of natural independence, i.e., tensor, free and Boolean independences. We also investigate relations between generating functions of moments and monotone cumulants. We find a natural extension of the Muraki formula, which describes the sum of monotone independent random variables, to the multivariate case.


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Takahiro Hasebe. Hayato Saigo. "Joint cumulants for natural independence." Electron. Commun. Probab. 16 491 - 506, 2011.


Accepted: 5 September 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1247.46052
MathSciNet: MR2836756
Digital Object Identifier: 10.1214/ECP.v16-1647

Primary: 46L53
Secondary: 05A18, 46L54


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