Central limit theorem for stationary sequences in the Hilbert space

Let $\xi_i$ be a centered strong stationary sequence in the separable Hilbert space $H$. We say that $\xi_i$ satisfy CLT if $S_n=n^{-1/2}(\xi_1+\dots+\xi_n)$ converges weakly to a Gaussian variable $\eta$, $\mathbf P\{\eta\in H\}=1$. We study some conditions on a mixing of a sequence $\xi_i$ for this sequence to satisfy CLT.