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：Stein的Gaussian进程近似法可以用来限制对光滑功能的预期值(c ⁇ adl ⁇ ag随机流程的$h$)与对精通目标随机流程的相同功能的预期值(Z$)之间的差别。 不幸的是,对光滑功能的类别来说,这很容易做到,这是非常有限的。在这里,我们证明是无限的维度高斯平滑的不平等,使功能类别能够大大扩大 -- -- 例如,利普西茨在统一指标方面的功能,以及任意事件指标 -- -- 以换取在界限上失去精确度。我们的不平等表现在光滑测试功能的界限上,预期的功能值为X$,这与典型的紧凑性标准密切相关,对Z$的预期值类似,对于设定的美元指标,美元/mathb{P}的概率(Z\ k ⁇ ta\setminus K ⁇ _\\\\\\\theta},目标进程接近$的界限。