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

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.”

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”.

DOI: 10.1214/10-AIHP379
Classification: 46L53,  46L54,  05A40
Keywords: monotone independence, cumulants, umbral calculus
@article{AIHPB_2011__47_4_1160_0,
     author = {Hasebe, Takahiro and Saigo, Hayato},
     title = {The monotone cumulants},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1160--1170},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {4},
     year = {2011},
     doi = {10.1214/10-AIHP379},
     zbl = {1273.46049},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/10-AIHP379/}
}
TY  - JOUR
AU  - Hasebe, Takahiro
AU  - Saigo, Hayato
TI  - The monotone cumulants
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
DA  - 2011///
SP  - 1160
EP  - 1170
VL  - 47
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/10-AIHP379/
UR  - https://zbmath.org/?q=an%3A1273.46049
UR  - https://doi.org/10.1214/10-AIHP379
DO  - 10.1214/10-AIHP379
LA  - en
ID  - AIHPB_2011__47_4_1160_0
ER  - 
%0 Journal Article
%A Hasebe, Takahiro
%A Saigo, Hayato
%T The monotone cumulants
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 1160-1170
%V 47
%N 4
%I Gauthier-Villars
%U https://doi.org/10.1214/10-AIHP379
%R 10.1214/10-AIHP379
%G en
%F AIHPB_2011__47_4_1160_0
Hasebe, Takahiro; Saigo, Hayato. The monotone cumulants. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 4, pp. 1160-1170. doi : 10.1214/10-AIHP379. http://www.numdam.org/articles/10.1214/10-AIHP379/

[1] A. C. R. Belton. A note on vacuum-adapted semimartingales and monotone independence. In Quantum Probability and Infinite Dimensional Analysis 105-114. QP-PQ: Quantum Probab. White Noise Anal. 18. World Sci. Publ., Hackensack, NJ, 2005. | MR

[2] A. C. R. Belton. On the path structure of a semimartingale arising from monotone probability theory. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 258-279. | Numdam | MR | Zbl

[3] A. Ben Ghorbal and M. Schürmann. Non-commutative notions of stochastic independence. Math. Proc. Comb. Phil. Soc. 133 (2002) 531-561. | MR | Zbl

[4] E. Di Nardo and D. Senato. Umbral nature of the Poisson random variables. In Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota 245-266. H. Crapo and D. Senato (Eds). Springer, Milan, 2001. | MR | Zbl

[5] K. L. Chung. A Course in Probability Theory. Brace & World, Harcourt, 1968. | MR | Zbl

[6] T. Hasebe and H. Saigo. Joint cumulants for natural independence. Available at arXiv:1005.3900v1. | MR

[7] F. Lehner. Cumulants in noncommutative probability theory I. Math. Z. 248 (2004) 67-100. | MR | Zbl

[8] R. Lenczewski and R. Sałapata. Discrete interpolation between monotone probability and free probability. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 77-106. | MR | Zbl

[9] R. Lenczewski and R. Sałapata. Noncommutative Brownian motions associated with Kesten distributions and related Poisson processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008) 351-375. | MR | Zbl

[10] N. Muraki. Monotonic convolution and monotonic Lévy-Hinčin formula. Preprint, 2000.

[11] N. Muraki. Monotonic independence, monotonic central limit theorem and monotonic law of small numbers. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001) 39-58. | MR | Zbl

[12] N. Muraki. The five independences as quasi-universal products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002) 113-134. | MR | Zbl

[13] N. Muraki. The five independences as natural products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003) 337-371. | MR | Zbl

[14] N. Obata. Notions of independence in quantum probability and spectral analysis of graphs. Amer. Math. Soc. Trans. 223 (2008) 115-136. | MR | Zbl

[15] G.-C. Rota and B. D. Taylor. The classical umbral calculus. SIAM J. Math. Anal. 25 (1994) 694-711. | MR | Zbl

[16] H. Saigo. A simple proof for monotone CLT. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010) 339-343. | MR | Zbl

[17] A. N. Shiryayev. Probability. Springer, New York, 1984. | MR

[18] R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 (1994) 611-628. | EuDML | MR | Zbl

[19] R. Speicher. On universal products. In Free Probability Theory 257-266. D. Voiculescu (Ed.). Fields Inst. Commun. 12. Amer. Math. Soc., Providence, RI, 1997. | MR | Zbl

[20] R. Speicher and R. Woroudi. Boolean convolution. In Free Probability Theory 267-280. D. Voiculescu (Ed.). Fields Inst. Commun. 12. Amer. Math. Soc., Providence, RI, 1997. | MR | Zbl

[21] D. Voiculescu. Symmetries of some reduced free product algebras. In Operator Algebras and Their Connections With Topology and Ergodic Theory 556-588. Lect. Notes in Math. 1132. Springer, Berlin, 1985. | MR | Zbl

[22] D. Voiculescu. Addition of certain non-commutative random variables. J. Funct. Anal. 66 (1986) 323-346. | MR | Zbl

Cited by Sources: