The monotone cumulants

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 47, Issue 4, 2011, Pages 1160-1170

  Hasebe, Takahiro ^a   Saigo, Hayato ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

Cumulants; Monotone independence; Umbral calculus

Indexed keywords

EID: 80054105536     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AIHP379     Document Type: Article

References (22)

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. MR2211883
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. MR2446323
3. A. Ben Ghorbal and M. Schürmann. Non-commutative notions of stochastic independence. Math. Proc. Comb. Phil. Soc. 133 (2002) 531-561. MR1919720
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. MR1854481
5. K. L. Chung. A Course in Probability Theory. Brace & World, Harcourt, 1968. MR0229268
6. T. Hasebe and H. Saigo. Joint cumulants for natural independence. Available at arXiv:1005.3900v1.
7. F. Lehner. Cumulants in noncommutative probability theory I. Math. Z. 248 (2004) 67-100. MR2092722
8. R. Lenczewski and R. Sa?apata. Discrete interpolation between monotone probability and free probability. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 77-106. MR2214503 (Pubitemid 43461825)
9. R. Lenczewski and R. Sa?apata. Noncommutative Brownian motions associated with Kesten distributions and related Poisson processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008) 351-375. MR2446514
10. N. Muraki. Monotonic convolution and monotonic Lévy-Hiňcin 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. MR1824472
12. N. Muraki. The five independences as quasi-universal products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002) 113-134. MR1895232
13. N. Muraki. The five independences as natural products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003) 337-371. MR2016316 (Pubitemid 37527972)
14. N. Obata. Notions of independence in quantum probability and spectral analysis of graphs. Amer. Math. Soc. Trans. 223 (2008) 115-136. MR2441422
15. G.-C. Rota and B. D. Taylor. The classical umbral calculus. SIAM J. Math. Anal. 25 (1994) 694-711. MR1266584
16. H. Saigo. A simple proof for monotone CLT. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010) 339-343.
17. A. N. Shiryayev. Probability. Springer, New York, 1984. MR0737192
18. R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 (1994) 611-628. MR1268597
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. MR1426844
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. MR1426845
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. MR0799593
22. D. Voiculescu. Addition of certain non-commutative random variables. J. Funct. Anal. 66 (1986) 323-346. MR0839105