51. Monotone, free, and boolean cumulants: a shuffle algebra approach
- Author
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Ebrahimi-Fard, Kurusch and Patras, Frederic
- Subjects
Mathematics - Combinatorics ,Mathematics - Functional Analysis ,Mathematics - Probability ,16T05, 16T10, 16T30, 46L53, 46L54 - 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., Comment: final version
- Published
- 2017
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