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On some generalization of the \(t\)-transformation. (English) Zbl 1214.46044

Bożejko, Marek (ed.) et al., Noncommutative harmonic analysis with applications to probability. II: Papers presented at the 11th workshop, Bȩdlewo, Poland, August 17–23, 2008. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-08-9/pbk). Banach Center Publications 89, 165-187 (2010).
Summary: Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation \(U^{\mathbb T}\) of probability measures on the real line \(\mathbb{R}\), which is another possible generalization of the \(t\)-transformation. Using that deformation, we define a new convolution by deformation of the free convolution. The central limit measure with respect to the \(\mathbb{T}\)-deformed free convolutions is still a Kesten measure, but the Poisson limit depends on the two parameters and is different from the Poisson measures for \((a,b)\)-deformations. We also show that the \(\mathbb{T}\)-deformed free convolution is different from the convolution obtained as the deformed conditionally free convolution of Bożejko, Leinert and Speicher. Thus, the \(\mathbb{T}\) deformed free convolution does not satisfy the Bożejko property.
For the entire collection see [Zbl 1189.46002].

MSC:

46L54 Free probability and free operator algebras
46L53 Noncommutative probability and statistics
60E10 Characteristic functions; other transforms
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