Rates of convergence of autocorrelation estimates for periodically correlated autoregressive Hilbertian processes

Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

“…(a) Theorems 3.1-3.2 extend the existing literature regarding the estimation of (lagged) covariance operators in several ways, see e.g. [1], [5], [16], [21] [26], [18], [19], [29], [30]. This is because the upper bounds in both Theorems are derived for lagged covariance operators of processes with arbitrary moments having values in arbitrary separable Hilbert spaces, and since the processes' 'outer dimension' m and simultaneously the lag h is allowed to grow w.r.t.…”

“…Probabilistic features of and estimators for lag-h-covariance operators C X;h of stationary processes X = (X k ) k∈Z with values in L 2 [0, 1], the space of measurable, square-Lebesgue integrable real valued functions with domain [0, 1], are widely studied for fixed lag h, see, e. g., [5], [19], [22], [34], [27]. Further, [39] developed covariance estimators in the space of continuous functions C[0, 1], [48] in tensor product Sobolev-Hilbert spaces, [33] for continuous surfaces, and [18], [1] for arbitrary separable Hilbert spaces. [34], [39], [18], [1] constrained their assertions to autoregressive (AR) processes, where [1] deduced the results for a random AR(1) operator.…”

“…Further, [39] developed covariance estimators in the space of continuous functions C[0, 1], [48] in tensor product Sobolev-Hilbert spaces, [33] for continuous surfaces, and [18], [1] for arbitrary separable Hilbert spaces. [34], [39], [18], [1] constrained their assertions to autoregressive (AR) processes, where [1] deduced the results for a random AR(1) operator. Thereby, [5], [19], [22], [1] utilized classical moment estimators, [27] estimated the integral kernels, in [18], [34] truncated spectral decompositions occured having estimated principle components, and [48] used operator regularized covariance estimators.…”

“…[34], [39], [18], [1] constrained their assertions to autoregressive (AR) processes, where [1] deduced the results for a random AR(1) operator. Thereby, [5], [19], [22], [1] utilized classical moment estimators, [27] estimated the integral kernels, in [18], [34] truncated spectral decompositions occured having estimated principle components, and [48] used operator regularized covariance estimators. Also, the limit distribution of the estimation errors of the lag-0-covariance operators was discussed in [26], [28].…”

“…Remarks 4.1. For extensive works dealing with functional AR(MA) processes, we refer to [5], [43] and also [16], [1], [7], [8], [34], [18] from a technical point of view, and [10], [25], [40] for methods combined with applications.…”