The characterizations of a semicircle law by the certain freeness in a C*-probability space |
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Authors: | Osamu Hiwatashi Masaru Nagisa Hiroaki Yoshida |
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Affiliation: | (1) Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33, Yayoi-Cho, Inage-Ku, Chiba 263, Japan, JP;(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 |
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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. Received: 12 February 1997 / Revised version: 16 June 1998 |
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Keywords: | Mathematics Subject Classification (1991): 46L50 60F05 62E10 |
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