On some generalization of the t-transformation

Anna Dorota Krystek

Banach Center Publications (2010)

  • Volume: 89, Issue: 1, page 165-187
  • ISSN: 0137-6934

Abstract

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Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation U 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.

How to cite

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Anna Dorota Krystek. "On some generalization of the t-transformation." Banach Center Publications 89.1 (2010): 165-187. <http://eudml.org/doc/282200>.

@article{AnnaDorotaKrystek2010,
abstract = {Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation $U^\{\}$ 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.},
author = {Anna Dorota Krystek},
journal = {Banach Center Publications},
keywords = {free convolution; conditionally free convolution; deformation; limit theorems; moment cumulant formulae},
language = {eng},
number = {1},
pages = {165-187},
title = {On some generalization of the t-transformation},
url = {http://eudml.org/doc/282200},
volume = {89},
year = {2010},
}

TY - JOUR
AU - Anna Dorota Krystek
TI - On some generalization of the t-transformation
JO - Banach Center Publications
PY - 2010
VL - 89
IS - 1
SP - 165
EP - 187
AB - Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation $U^{}$ 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.
LA - eng
KW - free convolution; conditionally free convolution; deformation; limit theorems; moment cumulant formulae
UR - http://eudml.org/doc/282200
ER -

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