Approximation of a diffusion process on an infinite-dimensional space using averaging of random shifts of general form

The purpose of this work is to study the dynamics of the diffusion process in an infinitedimensional space, in particular, its approximation using random walks. It is shown that for any distribution of vectors of a fairly general form, the process of averaging a random shift along these vectors converges to the evolution of the diffusion process. This result can also be considered as an analogue of central limit theorem for operator-valued functions on a Hilbert space.