Malliavin structure for conditionally independent random variables

On any denumerable product of probability spaces, we extend the discrete Malliavin structure for conditionally independent random variables. As a consequence, we obtain the chaos decomposition for functionals of conditionally independent random variables. We also show how to derive some concentration results in that framework. The Malliavin-Stein method yields Berry-Esseen bounds for U-Statistics of such random variables. It leads to quantitative statements of conditional limit theorems: Lyapunov's central limit theorem, De Jong's limit theorem for multilinear forms. The latter is related to the fourth moment phenomenon. The final application consists of obtaining the rates of normal approximation for subhypergraph counts in random exchangeable hypergraphs including the Erd{\"o}s-R{\'e}nyi hypergraph model. The estimator of subhypergraph counts is an example of homogeneous sums for which we derive a new decomposition that extends the Hoeffding decomposition.