Optimal rate for covariance operator estimators of functional autoregressive processes with random coefficients (Q1755118)

Let \((\Omega, \mathcal{A}, \mathbb{P})\) be a complete probability space, \(H\) be a real separable Hilbert space and \(\mathcal{L}(H)\) be the Banach space of continuous linear operators from \(H\) to \(H\) equipped with the usual operator norm. Let \(\rho_n\) be a sequence of measurable random operators defined on \(\Omega\) with values in \(\mathcal{L}(H)\). The authors of the paper consider the autoregressive equation \[ X_n=\rho_nX_{n-1}+\varepsilon_n, n\in\mathbb{Z}, \] where \(\varepsilon_n\), \(n\in\mathbb{Z}\) is an \(H\)-valued white noise. Properties of random \(H\)-valued sample \(X_1, X_2,\ldots, X_n\) are derived under suitable conditions on \(\rho_n\), \(\varepsilon_n\) and \(X_0\). In particular, the exponential bounds are derived for the empirical covariance operator of \(X_0\) in the Hilbert-Schmidt norm.