Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series

Takahiro Hasebe, Franz Lehner

Abstract

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 Möbius 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.

Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series

Abstract

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 Möbius 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.

Paper Structure

This paper contains 42 sections, 46 theorems, 247 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background: independence and cumulants
  3. Main objectives of the present paper
  4. Main results and structure of the paper
  5. Spreadability systems and independence
  6. Notation and terminology for noncommutative probability
  7. Spreadability systems
  8. Examples from universal products of linear functionals
  9. Tensor exchangeability system
  10. Free exchangeability system
  11. Boolean exchangeability system
  12. Monotone spreadability system
  13. Conditionally monotone spreadability system
  14. $V$-monotone spreadability system
  15. Spreadability systems with calculation rules
  16. ...and 27 more sections

Key Result

Proposition \oldthetheorem

Let $d \in \mathbb{N}$ and $\mathop{\mathrm{\odot}}\limits$ be a universal product in the category $\operatorname{AlgP}_d$ satisfying U1, U2 and U6. Let $(\mathcal{A}, \varphi^1,\varphi^2, \dots, \varphi^d)$ be an object in $\operatorname{AlgP}_d$ and $(\mathcal{A}_i)_{i \in I}$ be a family of subal

Figures (6)

  • Figure 1: Examples of partitions
  • Figure 2: The partitions $\mathop{\mathrm{\mathrm{\nu_{\max}}}}\nolimits(A)$ and $\mathop{\mathrm{\mathrm{\tilde{\nu}_{\max}}}}\nolimits(A)$ for $A=\{1,3,4,6,8,9\}$ (fat).
  • Figure 3: Examples of the order on $\mathcal{OP}_n$.
  • Figure 4: The poset $\mathcal{OP}_3$, ordered from outside to inside. The maximal element is in the center, the minimal elements are on the periphery.
  • Figure 5: Monotone partitions (upper row) and non-monotone partitions (lower row). The labeled numbers denote the order of blocks.
  • ...and 1 more figures

Theorems & Definitions (169)

  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • ...and 159 more