Abstract

Suppose ${\mathbb{V}}_{\nu }$ is the pseudo-variance function of the Cauchy–Stieltjes Kernel family ${\mathcal{K}}_{-}(\nu )$ generated by a non degenerate probability measure ν on the positive real line. We determine explicitly the law of large numbers μ for free multiplicative convolution given by Haagerup and Möller in terms of the pseudo-variance function ${\mathbb{V}}_{\nu }.$ Some examples of calculations of μ are given for probability measures ν of importance in free probability.

中文翻译：

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大数的显式自由乘法

摘要

认为${\mathbb{V}}_{\nu }$是 Cauchy–Stieltjes 内核系列的伪方差函数${\mathcal{钾}}_{-}(\nu )$由正实线上的非退化概率测度ν生成。我们明确地确定了Haagerup 和 Möller 根据伪方差函数给出的自由乘法卷积的大数定律μ${\mathbb{V}}_{\nu }.$对于自由概率中重要的概率度量ν，给出了μ的一些计算示例。