Estimation of Mean and Covariance Operator for Banach Space Valued Autoregressive Processes with Dependent Innovations

In this paper we study autoregressive processes of order 1 with values in a separable Banach space it B. Such ARB(1)-processes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(X_{n})}_{n \in \mathbb{Z}}$$\end{document} are defined by the recursion equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_n - m = T(x_{n-1}-m) + \epsilon_n, n \in \mathbb{Z}$$\end{document} where T : B → B is a bounded linear operator and m ∈ B. We analyze the asymptotic properties of the sample mean and of the sample covariance operator in case that the innovation process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\epsilon_{n})}_{n \in \mathbb{Z}}$$\end{document} is weakly dependent. This extends earlier results of Bosq (2000, 2002), who studied ARB(1)-processes with independent and orthogonal observations.