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Hinčin’s Theorem for Additive Free Convolutions of Tracial R-Diagonal \(*\)-Distributions

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Abstract

Hinčin proved that any limit law associated with a triangular array of uniformly infinitesimal random variables is infinitely divisible. Analogous results for the additive and multiplicative free convolution were proved by Bercovici, Belinschi and Pata. We prove an analogous result for the \(\boxplus _{RD}\) convolution of measures defined on the positive half-line. This is the convolution arising from the addition of \(*\)-free R-diagonal elements of a tracial, noncommutative probability space.

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Acknowledgements

The author was partially supported by a grant from the National Science Foundation. The author would like to thank Dr Hari Bercovici for his helpful advice on various technical issues examined in this Paper, and Dr Dan Voiculescu for his initiation of free probability theory. The author, however, bears full responsibility for the Paper.

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  1. Department of Mathematics, Indiana University, Bloomington, Rawles Hall, 831 East Third Street, Bloomington, IN, 47405, USA

    Cong Zhou

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  1. Cong Zhou
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Correspondence to Cong Zhou.

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Communicated by Hari Bercovici.

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This article is part of the topical collection “Infinite-dimensional Analysis and Non-commutative Theory” edited by Marek Bozejko, Palle Jorgensen and Yuri Kondratiev.

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Zhou, C. Hinčin’s Theorem for Additive Free Convolutions of Tracial R-Diagonal \(*\)-Distributions. Complex Anal. Oper. Theory 16, 1 (2022). https://doi.org/10.1007/s11785-021-01166-8

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  • DOI: https://doi.org/10.1007/s11785-021-01166-8

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