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Estimation of mean and covariance operator for Banach space valued autoregressive processes with dependent innovations. (English) Zbl 1079.62084

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.

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
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