Limit Theorems for the Hierarchy of Freeness

• Autorzy: Franz U. , Lenczewski R.
• Języki publikacji: EN

Abstrakty

EN

The central limit theorem, the invariance principle and the Poisson limit theorem for the hierarchy of freeness are studied. We show that for given m ϵ N the limit laws can be expressed in terms of non-crossing partitions of depth smaller than or equal to m. For A = C[x], we solve the associated moment problems and find explicitly the discrete limit measures.

Słowa kluczowe

EN

Poisson’s limit theorem; hierarchy of freeness; central limit theorem

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 1999
• Tom: Vol. 19, Fasc. 1
• Strony: 23--41
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Franz U.

• Institut de Recherche Mathématique Avancée, Université Louis Pasteur, Rue René Descartes, F-67084 Strasbourg Cedex, France

autor

Lenczewski R.

• Hugo Steinhaus Center for Stochastic Methods, Institute of Mathematics, Technical University of Wrocław, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

• [1] M. Bożejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175, No. 2 (1996), pp. 357-388.
• [2] U. Franz, R. Lenczewski and M. Schürmann, The GNS construction for the hierarchy of freeness, Preprint No. 9/98, Technical University of Wrocław, 1998.
• [3] R. Lenczewski, Unification of independence in quantum probability, Inf. Dim. Anal., Quant. Probab. & Rel. Top., Vol. 1, No. 3 (1998), pp. 383-405.
• [4] - A noncommutative limit theorem for homogeneous correlations, Studia Math. 129 (1998), pp. 225-252.
• [5] H. Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (1992), pp. 409-438.
• [6] A. Nica and R. Speicher (with an appendix by D. Voiculescu), On the multiplication of free N-tuples of noncommutative random variables, Amer. J. Math. 118 (1996), pp. 799-837.
• [7] M. Schürmann, Non-commutative probability on algebraic structures, in; Probability Measures on Groups and Related Structures, Vol. XI (Oberwolfach, 1994), World Scientific, River Edge, NJ, 1995, pp. 332-356.
• [8] – Direct sums of tensor products and non-commutative independence, J. Funct. Anal. 133 (1995), pp. 1-9.
• [9] R. Speicher, A new example of "independence" and "white noise", Probab. Theory Related Fields 84 (1990), pp. 141-159.
• [10] - and W. von Waldenfels, A general central limit theorem and invariance principle, in: Quantum Probability and Related Topics, Vol. IX, World Scientific, 1994, pp. 371-387.
• [11] D. V. Voiculescu, Symmetries of some reduced free product φ*-algebras, in: Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Math. 1132, Springer, Berlin 1985, pp. 556-588.
• [12] - Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), pp. 323-346.