Monotone, free, and boolean cumulants: a shuffle algebra approach

Abstract

The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.

arXiv commentfinal version

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Cited by in corpus (10)

• Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations
• Shuffle group laws. Applications in free probability
• Cumulant-cumulant relations in free probability theory from Magnus' expansion
• Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series
• Monotone Cumulant-Moment Formula and Schröder Trees
• Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
• Planar binary trees, noncrossing partitions and the operator-valued S-transform
• Tridendriform structures
• Algebraic groups in non-commutative probability theory revisited
• Conditionally monotone cumulants via shuffle algebra