论文标题
自由概率理论中的累积肿瘤关系来自Magnus的扩展
Cumulant-cumulant relations in free probability theory from Magnus' expansion
论文作者
论文摘要
矩和累积物之间的关系在经典和非交通概率理论中起着核心作用。后者允许几种与不同类型独立性相对应的累积剂族：自由，布尔和单调。最近研究了这些累积物之间的关系。在这项工作中，我们着重于用封闭的公式多元单调累积物来表达以自由和布尔累积液的形式表达的问题。在此过程中，我们介绍了有关非交叉分区的各种结构和统计数据。我们的方法基于累积功能的前代代数结构。累积物之间的关系是用前Lie Magnus膨胀以及由于A. murua引起的连续面包板 - 贝克 - 贝克 - 霍斯多夫公式的结果。
Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker-Campbell-Hausdorff formula due to A. Murua.
