Dehling, Herold; Sharipov, Olimjon Sh. Estimation of mean and covariance operator for Banach space valued autoregressive processes with dependent innovations. (English) Zbl 1079.62084 Stat. Inference Stoch. Process. 8, No. 2, 137-149 (2005). Summary: We study autoregressive processes of order 1 with values in a separable Banach space \(B\). Such ARB(1)-processes \((X_{n})_{n \in \mathbb Z}\) are defined by the recursion equation\[ X_n - m = T(X_{n-1}-m) + \varepsilon_n, \quad n \in \mathbb Z, \] where \(T : B\to B\) is a bounded linear operator and \(m\in B\). We analyze the asymptotic properties of the sample mean and of the sample covariance operator in the case that the innovation process \((\varepsilon_n)_{n \in \mathbb Z}\) is weakly dependent. This extends earlier results of D. Bosq [Linear processes in function spaces. Theory and applications. (2000; Zbl 0962.60004); Stat. Inference Stoch. Process. 5, 287–306 (2002; Zbl 1028.62070)], who studied ARB(1)-processes with independent and orthogonal observations. Cited in 14 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60B11 Probability theory on linear topological spaces 46N30 Applications of functional analysis in probability theory and statistics Keywords:mixing conditions; autoregressive process; sample mean; empirical covariance operator Citations:Zbl 0962.60004; Zbl 1028.62070 PDFBibTeX XMLCite \textit{H. Dehling} and \textit{O. Sh. Sharipov}, Stat. Inference Stoch. Process. 8, No. 2, 137--149 (2005; Zbl 1079.62084) Full Text: DOI References: This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.