Harmonic analysis for the bi-free partial S-transform |
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Authors: | Hao-Wei Huang Jiun-Chau Wang |
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Institution: | 1. Department of Applied Mathematics, National Sun Yat-sen University, No. 70, Lienhai Road, Kaohsiung 80424, Taiwan, ROC;2. Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan S7N 5E6, Canada |
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Abstract: | We develop an analytic machinery to study Voiculescu's bi-free partial S-transform and then use the results to characterize the multiplicative bi-free infinite divisibility. It is shown that the class of infinitely divisible distributions coincides with the class of limit distributions for products of bi-free pairs of left and right infinitesimal unitaries, where the pairs are not required to be identically distributed but all left variables are assumed to commute with all right variables. Furthermore, necessary and sufficient conditions for convergence to a given infinitely divisible distribution are found. |
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Keywords: | 46L54 Bi-free convolution Infinitely divisible distribution |
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