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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.

Wick polynomials and Wick products are studied in the context of non-commutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf algebraic approach to cumulants and Wick products in classical probability theory.

This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with many-body Feynman-Kac models. This perspective sheds a new light on the foundations and the mathematical analysis of this class of methods. A key consequence is the equivalence between the backward and ancestral particle Markov chain Monte Carlo methods, and Gibbs sampling of a many-body Feynman-Kac target distribution. Our approach also presents a new stochastic differential calculus based on geometric combinatorial techniques to derive explicit non-asymptotic Taylor type series of the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures with respect to the population size of the auxiliary particle sampler. These results provide sharp quan- titative estimates of the convergence properties of conditional particle Markov chain models with respect to the time horizon and the size of the systems. We illustrate the implication of these results with sharp estimates of the contraction coefficient and the Lyapunov exponent of conditional particle samplers, and explicit and non-asymptotic Lp-mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework developed in the article also allows the design of natural extensions to island (also called SMC2) type particle methodologies.

The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\WQSym$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\WQSym$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In particular, we compute their Adams operations and prove the existence of generalized Eulerian idempotents, that is, of a canonical left-inverse to the natural surjection map to their indecomposables, allowing for the combinatorial construction of free polynomial generators for these algebras., Comment: 18 pages

The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, de99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework.

Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann-Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme., Comment: typos corrected, references updated, accepted for publ. in Commun. Math. Phys

The Gerstenhaber and Schack cohomology comparison theorem asserts that there is a cochain equivalence between the Hochschild complex of a certain algebra and the usual singular cochain complex of a space. We show that this comparison theorem preserves the brace algebra structures. This result gives a structural reason for the recent results establishing fine topological structures on the Hochschild cohomology, and a simple way to derive them from the corresponding properties of cochain complexes., Comment: Revised version of "The bar construction as a Hopf algebra", Dec. 2001