The Lévy–Khintchine formula and Nica–Speicher property for deformations of the free convolution

Volume 78 / 2007

/Lukasz Jan Wojakowski Banach Center Publications 78 (2007), 309-314 MSC: Primary 46L53, 46L54; Secondary 60E10. DOI: 10.4064/bc78-0-23


We study deformations of the free convolution arising via invertible transformations of probability measures on the real line $T:\mu\mapsto T\mu$. We define new associative convolutions of measures by $$ \mu \mathbin{\mathbin{\boxplus }_T} \nu = T^{-1}(T\,\mu \mathbin{\boxplus } T\,\nu). $$ We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy–Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures $\mu$ have the Nica–Speicher property, that is, one can find their convolution power $\mu^{\mathbin{\mathbin{\boxplus }_T} s}$ for all $s\ge1$. This behaviour is similar to the free case, as in the original paper of Nica and Speicher [NS].


  • /Lukasz Jan WojakowskiMathematical Institute
    University of Wroc/law
    Pl. Grunwaldzki 2/4
    50-384 Wroc/law, Poland

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