The monotone cumulants
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1160-1170.

Dans cet article, nous définissons une notion de cumulants généralisés qui fournit un cadre commun pour les théories de probabilités commutatives, libres, booléennes et monotones. L'unicité des cumulants généralisés est vérifiée pour chacune de ces notions d'indépendance, qui par conséquent coincident avec les cumulants usuels dans les cadres commutatifs, libres et booléen. La façon dont nous définissons ces cumulants ne nécessite ni partition de réseaux ni fonction génératrice et donne un nouveau point de vue sur ces cumulants. Nous définissons des “cumulants monotones” et obtenons des preuves assez simples des théorémes de la limite centrale et de la distribution de Poisson dans le contexte des probabilités monotones. De plus, nous clarifions une structure combinatoire de la relation moments-cumulants à l'aide des “partitions monotones”.

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.”

DOI : 10.1214/10-AIHP379
Classification : 46L53, 46L54, 05A40
Mots-clés : monotone independence, cumulants, umbral calculus 
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Hasebe, Takahiro; Saigo, Hayato. The monotone cumulants. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1160-1170. doi : 10.1214/10-AIHP379. http://www.numdam.org/articles/10.1214/10-AIHP379/

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