GENERALIZED -TRANSFORMATIONS OF PROBABILITY MEASURES
AND DEFORMED CONVOLUTIONS
Anna Krystek
Hiroaki Yoshida
Abstract: In this paper, the generalized (two-parameterized) -transformations on probability
measures are introduced, in which the -transformation of Bożejko and Wysoczański can be
obtained as the special case, and the associated deformed convolutions are also investigated.
We see that the generalized -deformed free convolution can be realized as the conditionally
free convolution of Bożejko, Leinert, and Speicher. We also study another special
case of the generalized -deformed free convolution, which is called the -free
convolution, that gives an interpolation between the free and the Fermi convolutions.
2000 AMS Mathematics Subject Classification: Primary 46L53, 46L54; Secondary
60E10.
Key words and phrases: Convolution, conditionally free, moment-cumulant formula.