Superconvergence to the central limit and failure of the Cramér theorem for free random variables |
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Authors: | H Bercovici D Voiculescu |
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Institution: | (1) Mathematics Department, Indiana University, 47405 Bloomington, IN, USA;(2) Mathematics Department, University of California, 94720 Berkeley, CA, USA |
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Abstract: | 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.
The authors were partially supported by grants from the National Science Foundation |
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Keywords: | 46L50 60F05 60E07 |
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