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The monotone cumulants
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)
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Submitter: 長谷部 高広
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