Cumulant-cumulant relations in free probability theory from Magnus' expansion
- Others:
- Laboratoire Jean Alexandre Dieudonné (JAD) ; Université Nice Sophia Antipolis (1965 - 2019) (UNS) ; COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)
- ANR-20-CE40-0007,CARPLO,Combinatoire Algébrique, Renormalisation, Probabilités libres et Opérades(2020)
- ANR-21-CE48-0020,PAGCAP,Au delà du Permutoèdre et de l'Associaèdre : Géométrie, Combinatoire, Algèbre et Probabilité(2021)
- European Project: 670624,H2020,ERC-2014-ADG,DuaLL(2015)
Description
Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker-Campbell-Hausdorff formula due to A. Murua.
Abstract
International audience
Additional details
- URL
- https://hal.archives-ouvertes.fr/hal-03000717
- URN
- urn:oai:HAL:hal-03000717v1
- Origin repository
- UNICA