The monotone cumulants
Takahiro Hasebe,Hayato Saigo +1 more
TLDR
In this article, a notion of cumulants generalises is defined, which fournit un cadre commun for the theories of probabilites commutatives, libres, booleennes and monotones.Abstract:
Dans cet article, nous definissons une notion de cumulants generalises qui fournit un cadre commun pour les theories de probabilites commutatives, libres, booleennes et monotones. L’unicite des cumulants generalises est verifiee pour chacune de ces notions d’independance, qui par consequent coincident avec les cumulants usuels dans les cadres commutatifs, libres et booleen. La facon dont nous definissons ces cumulants ne necessite ni partition de reseaux ni fonction generatrice et donne un nouveau point de vue sur ces cumulants. Nous definissons des “cumulants monotones” et obtenons des preuves assez simples des theoremes de la limite centrale et de la distribution de Poisson dans le contexte des probabilites monotones. De plus, nous clarifions une structure combinatoire de la relation moments-cumulants a l’aide des “partitions monotones”.read more
Citations
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Relations between cumulants in noncommutative probability
TL;DR: In this paper, the authors express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations.
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Monotone Increment Processes, Classical Markov Processes, and Loewner Chains
TL;DR: In this article, it was shown that decreasing Loewner chains in the upper half-plane correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.
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Joint cumulants for natural independence
Takahiro Hasebe,Hayato Saigo +1 more
TL;DR: In this article, a unified treatment of joint cumulants is introduced for natural independence, which 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.
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Independences and Partial $R$-Transforms in Bi-Free Probability
TL;DR: In this article, the authors examined how various notions of independence in non-commutative probability theory arise in bi-free probability and established a Kac/Loeve Theorem for bi-freeness.
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Monotone, free, and boolean cumulants: a shuffle algebra approach
TL;DR: In this paper, the theory of cumulants is revisited in the Rota way by following a combinatorial Hopf algebra approach, and moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials.
References
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