Stein's method, smoothing and functional approximation

Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals h of a cadlag 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 provide 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 P(Z is an element of K theta \ K-theta) that the target process is close to the boundary of K.