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Contents of PMS, Vol. 33, Fasc. 2,
pages 341 - 352
 

AN EXAMPLE OF A BOOLEAN-FREE TYPE CENTRAL LIMIT THEOREM

Anna Kula
Janusz Wysoczański

Abstract: We construct a product of Hilbert spaces and associated product of operators, which generalizes the boolean and the free products and provides a model for new independence. The related Central Limit Theorem is proved.

2000 AMS Mathematics Subject Classification: Primary: 60F05, 60B12; Secondary: 05A18, 06A07, 46L53. Keywords and phrases: Free independence, boolean independence, noncommutative CLT, partially ordered sets.

References

[1]   M. Bożejko, Positive definite functions on the free group and the noncommutative Riesz product, Boll. Unione Mat. Ital. A (6) 5 (1) (1986), pp. 13–21.

[2]   J. H. Conway and R. Guy, The Book of Numbers, Springer, New York 1996.

[3]   J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., Oxford Sci. Publ., Clarendon Press, Oxford University Press, New York 1994.

[4]   R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. A Foundation for Computer Science, second edition, Addison-Wesley Publishing Company, Reading, MA, 1994.

[5]   A. Kula and J. Wysoczański, Noncommutative Brownian motions indexed by partially ordered sets, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (4) (2010), pp. 629–661.

[6]   N. Muraki, Monotonic independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001), pp. 39–58.

[7]   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 Math., Vol. 1132, Springer, Berlin 1985, pp. 556–588.

[8]   J. Wysoczański, Monotonic independence associated with partially ordered sets, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (1) (2007), pp. 17–41.

[9]   J. Wysoczański, bm-central limit theorems for positive definite real symmetric matrices, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (1) (2008), pp. 33–51.

[10]   J. Wysoczański, bm-independence and bm-central limit theorems associated with symmetric cones, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (3) (2010), pp. 461–488.

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