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

TL;DR

This paper establishes a Berry-Esseen type bound for the central limit theorem applied to non-causal linear processes in Hilbert spaces, under specific summability and boundedness conditions.

Contribution

It extends Berry-Esseen bounds to non-causal linear processes in Hilbert spaces with explicit conditions on operator coefficients and innovations.

Findings

Established Berry-Esseen bounds for Hilbert space-valued processes.

Derived convergence rates under summability and boundedness assumptions.

Provided conditions ensuring CLT rate accuracy in infinite-dimensional settings.

Abstract

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 $X_k=\sum_{j\in\mathbb{Z}}a_j(\varepsilon_{k-j})$ where $(\varepsilon_k)_{k\in\mathbb{Z}}$ 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 $\sum_{j\in\mathbb{Z}}\vert j\vert\|a_j\|_{\mathcal{L}(H)}<+\infty$ and $\varepsilon_0$ belongs to $L_H^{\infty}$.