We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

我们在高斯场的光滑泛函向量的分布与具有正定协方差矩阵的法向量的分布之间的凸距离上建立了明确的界限。我们的界限与 Nourdin 等人获得的界限相当。（Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010）用于（更平滑的）1-Wasserstein 距离，并且不涉及任何额外的对数因子。我们工作中使用的主要工具之一是对 Schulte 和 Yukich 最近获得的凸距离的递归估计（Electron J Probab 24(130):1-42, 2019）。我们在两种不同的情况下说明了我们的抽象结果：（i）我们证明了多个 Wiener-Itô 积分向量的定量多元四阶矩定理，