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The monotone cumulants

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Ann. I.H.P. Probab. stat47-4_1160-1170.pdf197.83 kBPDFView/Open
Please use this identifier to cite or link to this item:http://hdl.handle.net/2115/75991

Title: The monotone cumulants
Authors: Hasebe, Takahiro Browse this author →KAKEN DB
Saigo, Hayato Browse this author
Keywords: Monotone independence
Cumulants
Umbral calculus
Issue Date: 2011
Journal Title: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Volume: 47
Issue: 4
Start Page: 1160
End Page: 1170
Publisher DOI: 10.1214/10-AIHP379
Abstract: In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants. We define “monotone cumulants” in the sense of generalized cumulants and we obtain quite simple proofs of central limit theorem and Poisson’s law of small numbers in monotone probability theory. Moreover, we clarify a combinatorial structure of moment-cumulant formula with the use of “monotone partitions.”
Type: article
URI: http://hdl.handle.net/2115/75991
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: 長谷部 高広

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