Yannic Vargas - New formulas for cumulant-to-moment relations

This talk was part of the of the Master Class and Workshop on "Higher Structures Emerging from Renormalisation" held at the ESI November 8 - 19, 2021.

The study of relations between moments and cumulants play a central role in both classical and non-commutative probability theory. In the last decade, the work of Patras and Ebrahimi-Fard has provided several tools to use the group of characters on a combinatorial Hopf algebra H of "words on words", and its corresponding Lie algebra of infinitesimal characters, to study 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.