On the Free Convolution with a Free Multiplicative Analogue of the Normal Distribution

Ping Zhong ()
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Ping Zhong: Indiana University

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1354-1379

Abstract: Abstract We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $$\mathbb {C}$$ C with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane’s classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for probability measures on $$\mathbb {R}^+$$ R + . Finally, we describe the density of the free multiplicative analogue of the normal distributions as an example and prove unimodality and some symmetry properties of these measures.

Keywords: Free probability; Free Brownian motion; Multiplicative; Distribution; 46L54; 60F05 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-014-0556-x

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