Abstract
Recall from Chapter 1 that if \((\mathcal{A},\varphi )\) is a non-commutative probability space and \(\mathcal{A}_{1},\ldots,\mathcal{A}_{s}\) are subalgebras of \(\mathcal{A}\) which are free with respect to φ, then freeness gives us in principle a rule by which we can evaluate φ(a 1 a 2⋯a k ) for any alternating word in random variables a 1, a 2, …, a k . Thus we can in principle calculate all mixed moments for a system of free random variables. However, we do not yet have any concrete idea of the structure of this factorization rule. This situation will be greatly clarified by the introduction of free cumulants. Classical cumulants appeared in Chapter 1, where we saw that they are intimately connected with the combinatorial notion of set partitions. Our free cumulants will be linked in a similar way to the lattice of non-crossing set partitions; the latter were introduced in combinatorics by Kreweras [113]. We will motivate the appearance of free cumulants and non-crossing partition lattices in free probability theory by examining in detail a proof of the central limit theorem by the method of moments.
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References
N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis. Translated by N. Kemmer (Hafner Publishing Co., New York, 1965)
O. Arizmendi, T. Hasebe, F. Lehner, C. Vargas, Relations between cumulants in noncommutative probability. Adv. Math. 282, 56–92 (2015)
P. Biane, Processes with free increments. Math. Z. 227(1), 143–174 (1998)
P. Billingsley, Probability and Measure. Wiley Series in Probability and Mathematical Statistics, 3rd edn. (Wiley, New York, 1995). A Wiley-Interscience Publication
M. Bożejko, On Λ(p) sets with minimal constant in discrete noncommutative groups. Proc. Am. Math. Soc. 51(2), 407–412 (1975)
D.I. Cartwright, P.M. Soardi et al., Random walks on free products, quotients and amalgams. Nagoya Math. J. 102, 163–180 (1986)
P.L. Duren, Univalent Functions. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 259 (Springer, New York, 1983)
R.M. Friedrich, J. McKay, Homogeneous Lie groups and quantum probability. arXiv 1506.07089, June 2015
R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd edn. (Addison-Wesley Publishing Company, Reading, MA, 1994)
B. Krawczyk, R. Speicher, Combinatorics of free cumulants. J. Combin. Theory Ser. A 90(2), 267–292 (2000)
G. Kreweras, Sur les partitions non croisées d’un cycle. Discr. Math. 1(4), 333–350 (1972)
F. Lehner, Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems. Math. Zeitschrift 248(1), 67–100 (2004)
J.C. McLaughlin, Random walks and convolution operators on free products. Ph.D. thesis, New York University, 1986
A. Nica, R-transforms of free joint distributions and non-crossing partitions. J. Funct. Anal. 135(2), 271–296 (1996)
A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335 (Cambridge University Press, Cambridge, 2006)
M. Schürmann, S. Voß, Schoenberg correspondence on dual groups. Commun. Math. Phys. 328(2), 849–865 (2014)
R. Speicher, A new example of “independence” and “white noise”. Probab. Theory Relat. Fields 84(2), 141–159 (1990)
R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298(1), 611–628 (1994)
R.P. Stanley, Enumerative Combinatorics, vol. 1 (Cambridge University Press, Cambridge, 1997)
D. Voiculescu, Symmetries of some reduced free product C ∗-algebras, in Operator Algebras and Their Connections with Topology and Ergodic Theory (Buşteni, 1983). Lecture Notes in Mathematics, vol. 1132 (Springer, Berlin, 1985), pp. 556–588
D. Voiculescu, Addition of certain noncommuting random variables. J. Funct. Anal. 66(3), 323–346 (1986)
D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. I. Commun. Math. Phys. 155(1), 71–92 (1993)
D. Voiculescu, The coalgebra of the free difference quotient and free probability. Int. Math. Res. Not. 2000(2), 79–106 (2000)
W. Woess, Nearest neighbour random walks on free products of discrete groups. Boll. Unione Mat. Ital. VI. Ser. B 5, 961–982 (1986)
W. Woess, Random Walks on Infinite Graphs and Groups, vol. 138 (Cambridge University Press, Cambridge, 2000)
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Mingo, J.A., Speicher, R. (2017). The Free Central Limit Theorem and Free Cumulants. In: Free Probability and Random Matrices. Fields Institute Monographs, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6942-5_2
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