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Superconvergence to the central limit and failure of the Cramér theorem for free random variables
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  • Published: June 1995

Superconvergence to the central limit and failure of the Cramér theorem for free random variables

  • H. Bercovici1 &
  • D. Voiculescu2 

Probability Theory and Related Fields volume 103, pages 215–222 (1995)Cite this article

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Summary

We show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected. Thus, the distributions which tend to the semicircle become absolutely continuous in finite time, and the densities converge in a very strong sense. We also show that the semicircle law is the free convolution of laws which are not semicircular, thus proving that Cramér's classical result for the normal distribution does not have a free counterpart.

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Author information

Authors and Affiliations

  1. Mathematics Department, Indiana University, 47405, Bloomington, IN, USA

    H. Bercovici

  2. Mathematics Department, University of California, 94720, Berkeley, CA, USA

    D. Voiculescu

Authors
  1. H. Bercovici
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  2. D. Voiculescu
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Additional information

The authors were partially supported by grants from the National Science Foundation

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Cite this article

Bercovici, H., Voiculescu, D. Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Th. Rel. Fields 103, 215–222 (1995). https://doi.org/10.1007/BF01204215

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  • Received: 25 May 1994

  • Revised: 12 April 1995

  • Issue Date: June 1995

  • DOI: https://doi.org/10.1007/BF01204215

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Mathematics Subject Classifcation (1991)

  • 46L50
  • 60F05
  • 60E07
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