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The characterizations of a semicircle law by the certain freeness in a C*-probability space
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  • Published: February 1999

The characterizations of a semicircle law by the certain freeness in a C*-probability space

  • Osamu Hiwatashi1,
  • Masaru Nagisa1 &
  • Hiroaki Yoshida2 

Probability Theory and Related Fields volume 113, pages 115–133 (1999)Cite this article

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  • 6 Citations

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Abstract.

In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper, we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables, which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system. Our proof is more direct and straightforward one.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33, Yayoi-Cho, Inage-Ku, Chiba 263, Japan, , , , , , JP

    Osamu Hiwatashi & Masaru Nagisa

  2. Department of Information Sciences, Faculty of Science, Ochanomizu University, 2-1-1, Otsuka, Bunkyou, Tokyo 112, Japan. e-mail: yoashida@is.ocha.ac.jp, , , , , , JP

    Hiroaki Yoshida

Authors
  1. Osamu Hiwatashi
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  2. Masaru Nagisa
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  3. Hiroaki Yoshida
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Received: 12 February 1997 / Revised version: 16 June 1998

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Hiwatashi, O., Nagisa, M. & Yoshida, H. The characterizations of a semicircle law by the certain freeness in a C*-probability space. Probab Theory Relat Fields 113, 115–133 (1999). https://doi.org/10.1007/s004400050204

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  • Issue Date: February 1999

  • DOI: https://doi.org/10.1007/s004400050204

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  • Mathematics Subject Classification (1991): 46L50
  • 60F05
  • 62E10
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