Combinatorial Hopf algebras in noncommutative probabilility

Abstract

We prove that the generalized moment-cumulant relations introduced in [arXiv:1711.00219] are given by the action of the Eulerian idempotents on the Solomon-Tits algebras, whose direct sum builds up the Hopf algebra of Word Quasi-Symmetric Functions $\WQSym$. We prove $t$-analogues of these identities (in which the coefficient of $t$ gives back the original version), and a similar $t$-analogue of Goldberg's formula for the coefficients of the Hausdorff series. This amounts to the determination of the action of all the Eulerian idempotents on a product of exponentials.

44 pages

References in corpus (1)

• Cumulants in noncommutative probability II. Generalized Gaussian random variables

Cited by in corpus (1)

• Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series