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## 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.

#### Authors

• Anna KulaInstitute of Mathematics, Jagiellonian University
ul. S. Łojasiewicza 6, 30-348 Kraków, Poland
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