累积量,可扩展性和Campbell-Baker-Hausdorff系列

IF 0.9 3区 数学 Q2 MATHEMATICS
Takahiro Hasebe, Franz Lehner
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引用次数: 6

摘要

本文章由计算机程序翻译,如有差异,请以英文原文为准。
Cumulants, spreadability and the Campbell–Baker–Hausdorff series
We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and M\"obius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faa di Bruno's theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell-Baker-Hausdorff series in this context. In a final section we exhibit an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients.
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来源期刊
Documenta Mathematica
Documenta Mathematica 数学-数学
CiteScore
1.60
自引率
11.10%
发文量
0
审稿时长
>12 weeks
期刊介绍: DOCUMENTA MATHEMATICA is open to all mathematical fields und internationally oriented Documenta Mathematica publishes excellent and carefully refereed articles of general interest, which preferably should rely only on refereed sources and references.
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