Electronic Communications in Probability

Joint cumulants for natural independence

Takahiro Hasebe and Hayato Saigo

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Abstract

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.

Article information

Source
Electron. Commun. Probab. Volume 16 (2011), paper no. 44, 491-506.

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

Permanent link to this document
http://projecteuclid.org/euclid.ecp/1465262000

Digital Object Identifier
doi:10.1214/ECP.v16-1647

Mathematical Reviews number (MathSciNet)
MR2836756

Zentralblatt MATH identifier
1247.46052

Subjects
Primary: 46L53: Noncommutative probability and statistics
Secondary: 46L54: Free probability and free operator algebras 05A18: Partitions of sets

Keywords
Natural independence cumulants non-commutative probability monotone independence

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Hasebe, Takahiro; Saigo, Hayato. Joint cumulants for natural independence. Electron. Commun. Probab. 16 (2011), paper no. 44, 491--506. doi:10.1214/ECP.v16-1647. http://projecteuclid.org/euclid.ecp/1465262000.


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