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The Lebesgue decomposition of the free additive convolution of two probability distributions
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  • Published: 19 September 2007

The Lebesgue decomposition of the free additive convolution of two probability distributions

  • Serban Teodor Belinschi1 nAff2 

Probability Theory and Related Fields volume 142, pages 125–150 (2008)Cite this article

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Abstract

We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on \({\mathbb{R}}\) only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper.

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Author notes
  1. Serban Teodor Belinschi

    Present address: Department of Pure Mathematics, University of Waterloo, 200 University Street West, Waterloo, ON, N2L 3G1, Canada

Authors and Affiliations

  1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania

    Serban Teodor Belinschi

Authors
  1. Serban Teodor Belinschi
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Correspondence to Serban Teodor Belinschi.

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Cite this article

Belinschi, S.T. The Lebesgue decomposition of the free additive convolution of two probability distributions. Probab. Theory Relat. Fields 142, 125–150 (2008). https://doi.org/10.1007/s00440-007-0100-3

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  • Received: 30 September 2006

  • Revised: 13 July 2007

  • Published: 19 September 2007

  • Issue Date: September 2008

  • DOI: https://doi.org/10.1007/s00440-007-0100-3

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Mathematics Subject Classification (2000)

  • Primary 46L54
  • Secondary 30D40
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