On a filtering problem for vector-valued sequences. (English. Russian original) Zbl 0835.60038

Moklyachuk, M. P.

Theory Probab. Math. Stat. 47, 107-118 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 104-118 (1992).

Summary: The problem considered consists in optimal linear estimation of the transformation \(A \xi = \sum^\infty_{j = 0} \langle a(j), \xi (-j) \rangle\) of a stationary sequence \(\xi(j)\) with values in a Hilbert space from observations of the sequence \(\xi (k) + \eta (k)\) for \(k \leq 0\). Formulas are derived for computing the mean-square error and the spectral characteristic of the optimal estimate of the transformation \(A \xi\). The minimax spectral characteristics of the optimal estimate of \(A \xi\) and the least favorable spectral densities \(f^0 (\lambda)\) and \(g^0 (\lambda)\) are found for various classes, \(D_f\) and \(D_g\), of spectral densities.

MSC:

60G35	Signal detection and filtering (aspects of stochastic processes)
62M20	Inference from stochastic processes and prediction
93E10	Estimation and detection in stochastic control theory
93E11	Filtering in stochastic control theory

Keywords:

optimal linear estimation; mean-square error; spectral characteristic; spectral densities