It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of samples Gaussian process X controlled by certain nuclear dominance condition. However, it less clear how to identify “small” set functions (not necessarily vector space) contains samples. This article presents general approach for identifying such sets. We use scaled RKHSs, which can be viewed as generalisatio...

We establish the optimal rates of strong approximation by Wiener process for vector-valued cumulative processes. The Komlós–Major–Tusnády bounds are given both in case when exponential moments exist and power case. Applications to invariance principle stopped sums birth death processes provided. As a tool we use maximal inequality over random intervals which is independent interest.