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The Urbanik generalized convolutions in the non-commutative probability and a forgotten method of constructing generalized convolution

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Abstract

The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probability, two operations: the q-convolution and the (q,1)-convolution satisfy the Urbanik’s conditions for a generalized convolution, interpreted on the set of moment sequences. The weak stability reveals the relation between two operations.

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Acknowledgements

The second author (AK) was partially supported by the MNiSW research grant N201 36 44 36.

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Authors and Affiliations

  1. Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    BARBARA JASIULIS-GOŁDYN

  2. Institute of Mathematics, Jagiellonian University, ul. S. Łojasiewicza 6, 30-348, Kraków, Poland

    ANNA KULA

Authors
  1. BARBARA JASIULIS-GOŁDYN
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  2. ANNA KULA
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Correspondence to BARBARA JASIULIS-GOŁDYN.

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JASIULIS-GOŁDYN, B., KULA, A. The Urbanik generalized convolutions in the non-commutative probability and a forgotten method of constructing generalized convolution. Proc Math Sci 122, 437–458 (2012). https://doi.org/10.1007/s12044-012-0085-4

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  • DOI: https://doi.org/10.1007/s12044-012-0085-4

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