A CONNECTION BETWEEN FREE AND CLASSICAL INFINITE DIVISIBILITY
A CONNECTION BETWEEN FREE AND CLASSICAL INFINITE DIVISIBILITY
复制标题
自由和经典无限可分性之间的联系
DOI：
10.1142/s0219025704001773
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
S. Thorbjørnsen
中科院分区：
文献类型：
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作者：
O. Barndorff;S. Thorbjørnsen
在本文中，我们继续我们在参考文献2 - 4中发起的关于经典概率和自由概率中无穷可分概率测度类之间联系的研究。我们表明，任何自由无穷可分概率测度的自由累积量变换等于某一经典无穷可分概率测度的经典累积量变换，并且我们给出了后者测度的几种特征描述，包括一种关于随机积分的解释。此外，我们找到了贝尔科维奇 - 帕塔双射的另一种定义，它直接从经典累积量变换到自由累积量变换，而不经过列维 - 辛钦表示（分别为经典的和自由的）。
In this paper we continue our studies, initiated in Refs. 2–4, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici–Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Levy–Khintchine representations (classical and free, respectively).