Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation of probability measures on the real line 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 -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)-deformation. We also show that the -deformed free convolution is different from the convolution obtained as the deformed conditionally free convolution of Bożejko, Leinert and Speicher. Thus the does not satisfy the Bożejko property.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-10,
author = {Anna Dorota Krystek},
title = {On some generalization of the t-transformation},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {165-187},
zbl = {1214.46044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-10}
}
Anna Dorota Krystek. On some generalization of the t-transformation. Banach Center Publications, Tome 89 (2010) pp. 165-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-10/