The monotone cumulants

Takahiro Hasebe; Hayato Saigo

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 4, page 1160-1170
  • ISSN: 0246-0203

Abstract

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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.”

How to cite

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Hasebe, Takahiro, and Saigo, Hayato. "The monotone cumulants." Annales de l'I.H.P. Probabilités et statistiques 47.4 (2011): 1160-1170. <http://eudml.org/doc/240114>.

@article{Hasebe2011,
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.”},
author = {Hasebe, Takahiro, Saigo, Hayato},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {monotone independence; cumulants; umbral calculus},
language = {eng},
number = {4},
pages = {1160-1170},
publisher = {Gauthier-Villars},
title = {The monotone cumulants},
url = {http://eudml.org/doc/240114},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Hasebe, Takahiro
AU - Saigo, Hayato
TI - The monotone cumulants
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 4
SP - 1160
EP - 1170
AB - 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.”
LA - eng
KW - monotone independence; cumulants; umbral calculus
UR - http://eudml.org/doc/240114
ER -

References

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