A filtration of transformations of random sequences (Q1819815)

Let \(\xi\) (n), \(\eta\) (n) be uncorrelated, stationary random sequences with values in a separable Hilbert space \({\mathcal H}\). Consider the transformation \(A_{\xi}\) of the sequence \(\xi\) defined as \(A_{\xi}=\sum^{\infty}_{j=0}<\xi (j),a(j)>\) where \(<\cdot,\cdot >\) is the inner product in \({\mathcal H}\). The optimal linear predictor of \(A_{\xi}\) based on observations of the sequence \(\xi (n)+\eta (n)\) requires knowledge of the spectral densities of \(\xi (n)+\eta (n)\) requires knowledge of the spectral densities of \(\xi\) (n) and \(\eta\) (n). The author assumes that they exist but are not known. First he considers the problem of estimating \(A_ N\xi =\sum^{N}_{j=0}<\xi (j),q(j)>\) by \(\hat A_ N\xi\) based on the noisy observations \(\xi (n)+\eta (n)\), \(n=-1,-2,... \). The minimax estimation error is derived. Next the limiting values of the resulting expressions are calculated for \(N\to \infty\) and minimax prediction error for \(\hat A_{\xi}\) is derived.