Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space

Let $H$ be a real separable Hilbert space and $(a_k{)}_{k\in \mathbb{Z}}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\mathbb{Z}$ by $X_k={\sum }_{j\in \mathbb{Z}}{a}_j({\epsilon }_{k-j})$ where $({\epsilon }_k{)}_{k\in \mathbb{Z}}$ is a sequence of i.i.d. centered $H$-valued random variables. We investigate the rate of convergence in the CLT for $X$ and in particular we obtain the usual Berry-Esseen’s bound provided that ${\sum }_{j\in \mathbb{Z}}\mathrm{\mid }j\mathrm{\mid }\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{\mathcal{L}(H)}<+\mathrm{\infty }$ and ${\epsilon }_0$ belongs to $L_H^{\mathrm{\infty }}$.