New bounds for normal approximation on product spaces with applications to monochromatic edges, random sums and an infinite de Jong CLT

Abstract: We extend the Malliavin theory for $L^2$-functionals on product probability spaces that has recently been developed by Decreusefond and Halconruy (2019) and by Duerinckx (2021), by characterizing the domains and investigating the actions of the three Malliavin operators in terms of the infinite Hoeffding decomposition in $L^2$, which we identify as the natural analogue of the famous Wiener-It^{o} chaos decomposition on Gaussian and Poisson spaces. We further combine this theory with Stein's method for normal approximation in order to provide three different types of abstract Berry-Esseen and Wasserstein bounds: a) Malliavin-Stein bounds involving the Malliavin gradient $D$ and the pseudo-inverse of the Ornstein-Uhlenbeck generator $L$, b) bounds featuring the carr\'{e}-du-champ operator $\Gamma$ and c) bounds making use of a Clark-Ocone type integration-by-parts formula. To demonstrate the flexibility of these abstract bounds, we derive quantitative central limit theorems for the number of monochromatic edges in a uniform random coloring of a graph sequence as well as for random sums and prove an infinite version of the quantitative de Jong CLT that has recently been proved by G. Peccati and the author (2017) and by the author (2023). As a further theoretical application, we deduce new abstract Berry-Esseen and Wasserstein bounds for functionals of a general independent Rademacher sequence.