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On minimax filtration of vector processes

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Abstract

We study the problem of optimal linear estimation of the transformation\(A\xi = \smallint _0^\infty< a(t), \xi ( - t) > dt\) of a stationary random process ξ(t) with values in a Hilbert space by observations of the process ξ(t) + η(t) fort⩽0. We obtain relations for computing the error and the spectral characteristic of the optimal linear estimate of the transformationAξ for given spectral densities of the processes ξ(t) and η(t). The minimax spectral characteristics and the least favorable spectral densities are obtained for various classes of densities.

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References

  1. Yu. A. Rozanov,Theory of Renewal Processes [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  2. G. Kallianpur and V. Mandrekar, “Multiplicity and representation theory of purely nondeterministic stochastic processes,”Teor: Veroyatn. Primen.,10, No. 4, 614–644 (1965).

    Google Scholar 

  3. S. A. Kassam and V. H. Poor, “Robust techniques for signal processing: A survey,”Proc. IEEE,73, No 3, 433–481 (1985).

    Google Scholar 

  4. J. Franke, “Minimax-robust prediction of discrete time series,”Z. Wahrscheinlichkeitstheorie,68, No 2, 337–364 (1985).

    Google Scholar 

  5. J. Franke and V. H. Poor, “Minimax-robust filtering and finite-length robust predictors,”Lecture Notes Statist,26, 87–126 (1984).

    Google Scholar 

  6. M. P. Moklyachuk, “Estimation of linear functionals of stationary stochastic processes and a two-person zero-sum game,”Stanford Univ. Techn. Report, No. 169, (1981).

  7. M. P. Moklyachuk, “On a problem of the theory of games and extrapolation of random processes with values in a Hilbert space,”Teor. Ver. Mat. Statist., Issue 24, 107–114 (1981).

    Google Scholar 

  8. M. P. Moklyachuk, “Minimax filtration of linear transformations of stationary processes,”Teor. Ver. Mat. Statist., Issue 44, 96–105 (1991).

    Google Scholar 

  9. B. N. Pshenichnyi,Necessary Conditions of Extremum [in Russian], Nauka, Moscow (1982).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    M. P. Moklyachuk

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  1. M. P. Moklyachuk
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 389–397, March, 1993.

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Moklyachuk, M.P. On minimax filtration of vector processes. Ukr Math J 45, 414–423 (1993). https://doi.org/10.1007/BF01061014

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  • DOI: https://doi.org/10.1007/BF01061014

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