In this paper, we develop the notion of free-Boolean independence in an amalgamated setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence with amalgamation. We also provide a characterization of free-Boolean independence by conditions in terms of mixed moments. In addition, we study free-Boolean independence over a [Formula: see text]-algebra and prove a positivity property. A central limit law for our free-Boolean independence with amalgamation is also studied.

中文翻译：

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具有合并的自由布尔独立性

在本文中，我们在合并环境中开发了自由布尔独立性的概念。我们构造了自由布尔累积量，并表明混合的自由布尔累积量的消失等同于我们的自由布尔独立性与合并。我们还根据混合矩的条件提供了对自由布尔独立性的表征。此外，我们研究了 [公式：参见文本]-代数上的自由布尔独立性，并证明了正性。我们还研究了具有合并的自由布尔独立性的中心极限定律。