# Contributions to functional inequalities and limit theorems on the configuration space

Abstract : We present functional inequalities and limit theorems for point processes. We prove a modified logarithmic Sobolev inequalities, a Stein inequality and a exact fourth moment theorem for a large class of point processes including mixed binomial processes and Poisson point processes. The proofs of these inequalities are inspired by the Malliavin-Stein approach and the $Gamma$-calculus of Bakry-Emery. The implementation of these techniques requires a development of a stochastic analysis for point processes. As point processes are essentially discrete, we design a theory to study non-diffusive random objects. For Poisson point processes, we extend the Stein inequality to study stable convergence with respect to limits that are conditionally Gaussian. Applications to Poisson approximations of Gaussian processes and random geometry are given. We discuss transport inequalities for mixed binomial processes and their consequences in terms of concentration of measure. On a generic metric measured space, we present a refinement of the notion of concentration of measure that takes into account the parallel enlargement of distinct sets. We link this notion of improved concentration with the eigenvalues of the metric Laplacian and with a version of the Ricci curvature based on multi-marginal optimal transport
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Ronan Herry. Contributions to functional inequalities and limit theorems on the configuration space. General Mathematics [math.GM]. Université Paris-Est; Université du Luxembourg, 2018. English. ⟨NNT : 2018PESC1134⟩. ⟨tel-02085829⟩

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