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Article Dans Une Revue Foundations of Computational Mathematics Année : 2022

Cumulant-cumulant relations in free probability theory from Magnus' expansion

A. Celestino
  • Fonction : Auteur
K. Ebrahimi-Fard
  • Fonction : Auteur
D. Perales Anaya
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-03000717 , version 1 (12-11-2020)

Identifiants

Citer

Frédéric Patras, A. Celestino, K. Ebrahimi-Fard, D. Perales Anaya. Cumulant-cumulant relations in free probability theory from Magnus' expansion. Foundations of Computational Mathematics, 2022, 22 (3), pp.733-755. ⟨10.1007/s10208-021-09512-0⟩. ⟨hal-03000717⟩
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