Weak convergence of probability measures in the spaces of continuously differentiable functions

A criterion for weak convergence of probability measures in the spaces of continuously dif-ferentiable functions $C^p(X)$, $X\subset\mathbb{R}^k$, is proved. As a consequence spectral conditions of relative compactness for a sequence of homogeneous fields and a central limit theorem in $C^p(X)$ are established.