Central limit theorems in Hölder topologies for Banach space valued random fields

For rather general moduli of smoothness $\rho$, such as $\rho(h)=h^\alpha \log^\beta (c/h)$, we consider the Hölder spaces $H_{\rho}(B)$ of functions $[0,1]^d \to B$, where $B$ is a separable Banach space. Using isomorphism between $H_{\rho}(B)$ and some sequence Banach space we follow a very natural way to study, in terms of second differences, the central limit theorem for independent identically distributed sequences of random elements in $H_{\rho}(B)$.