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

Mohamed El Machkouri (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

Let $H$ be a real separable Hilbert space and $(a_k)_{k\in\Z}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\Z$ by $X_k=\sum_{j\in\Z}a_j(\varepsilon_{k-j})$ where $(\varepsilon_k)_{k\in\Z}$ 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 $\sum_{j\in\Z}\vert j\vert\|a_j\|_{\L(H)}<+\infty$ and $\varepsilon_0$ belongs to $L_H^{\infty}$.

Mots clés: Central limit theorem; Berry-Esseen bound; Linear process; Hilbert space

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Mohamed El Machkouri. Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space. 2010. ⟨hal-00488676⟩