A limit theorem for the $\boldsymbol q$-convolution
Volume 96 / 2011
Banach Center Publications 96 (2011), 245-255
MSC: Primary: 60F05, 28D05; Secondary: 42A85, 44A60.
DOI: 10.4064/bc96-0-16
Abstract
The $q$-convolution is a measure-preserving transformation which originates from non-commutative probability, but can also be treated as a one-parameter deformation of the classical convolution. We show that its commutative aspect is further certified by the fact that the $q$-convolution satisfies all of the conditions of the generalized convolution (in the sense of Urbanik). The last condition of Urbanik's definition, the law of large numbers, is the crucial part to be proved and the non-commutative probability techniques are used.