Algebraic combinatorics approach to moment-to-cumulant relations in free probability
– 2.22 und online
Yannic Vargas (University of Potsdam)
The study of relations between moments and cumulants plays a central role in both classical and non-commutative probability theory. In the last decade, the work of Patras and Ebrahimi-Fard provided several tools related to the group of characters on a combinatorial Hopf algebra H of "words on words", and its corresponding Lie algebra of infinitesimal characters. This enables the study of distinct families of cumulants corresponding to different types of independences: free, boolean and monotone. We discuss several formulas for the (known) free-to-moment and boolean-to-moment relations, obtained from the antipode of H. Also, using a weighted Möbius inversion, we deduce a new relation of monotone cumulants in terms of moments.
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