Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals $h$ of a c\`adl\`ag random process $X$ of interest and the expectations of the same functionals of a well understood target random process $Z$ with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we prove an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded -- examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events -- in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of $X$ that is closely related to classical tightness criteria, a similar expectation for $Z$, and, for the indicator of a set $K$, the probability $\mathbb{P}(Z \in K^\theta \setminus K^{-\theta})$ that the target process is close to the boundary of $K$.

翻译：斯坦因方法用于高斯过程逼近时，可界定感兴趣的上半连续随机过程$X$的光滑泛函$h$的期望与具有连续路径的已知目标随机过程$Z$的同类泛函期望之间的差异。然而，可轻松应用该方法的泛函类别极为有限。本文证明了一个无穷维高斯平滑不等式，该不等式使得泛函类别得以大幅扩展——例如一致度量下的利普希茨泛函以及任意事件的示性函数——但需以边界精度的损失为代价。我们的不等式以光滑检验函数边界、与经典紧性准则密切相关的$X$泛函的期望、$Z$的类似期望，以及对于集合$K$的示性函数，目标过程接近$K$边界的概率$\mathbb{P}(Z \in K^\theta \setminus K^{-\theta})$作为表达形式。