An Improved Second Order Poincar\'e Inequality for Functionals of Gaussian Fields

We present an improved version of the second order Gaussian Poincar'e inequality, firstly introduced in Chatterjee (2009) and Nourdin, Peccati and Reinert (2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative CLTs for non-linear functionals of stationary Gaussian fields related to the Breuer-Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.