Let ${{xi}_k,k{in}{mathbb{Z}}}$ be an associated H-valued random variables with $E{xi}_k$ = 0, $E{parallel}{xi}_k{parallel}$ < ${infty}$ and $E{parallel}{xi}_k{parallel}^2$ < ${infty}$ and {$a_k,k{in}{mathbb{Z}}$} a sequence of bounded linear operators such that ${sum}^{infty}_{j=0}j{parallel}a_j{parallel}_{L(H)}$ < ${infty}$. We define the sationary Hilbert space process $X_k={sum}^{infty}_{j=0}a_j{xi}_{k-j}$ and prove that $n^{-1}{sum}^n_{k=1}X_k$ converges to zero.