Filtration of linear functionals of periodically correlated sequences

Authors:
I. I. Dubovets′ka and M. P. Moklyachuk

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **86** (2012).

Journal:
Theor. Probability and Math. Statist. **86** (2013), 51-64

MSC (2010):
Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11

DOI:
https://doi.org/10.1090/S0094-9000-2013-00888-6

Published electronically:
August 20, 2013

MathSciNet review:
2986449

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of the optimal estimation is considered for the linear functional

**1.**W. R. Bennett,*Statistics of regenerative digital transmission*, Bell Syst. Tech.**37**(1958), 1501-1542. MR**0102138 (21:932)****2.**J. Franke,*On the robust prediction and interpolation of time series in the presence of correlated noise*, J. Time Series Anal.**5**(1984), 227-244. MR**782077 (86i:62192)****3.**J. Franke,*Minimax robust prediction of discrete time series*, Z. Wahrsch. Verw. Gebiete**68**(1985), 337-364. MR**771471 (86f:62164)****4.**J. Franke and H. V. Poor,*Minimax-robust filtering and finite-length robust predictors*, Robust and Nonlinear Time Series Analysis, Lecture Notes in Statistics, vol. 26, Springer-Verlag, 1984. MR**786305 (86i:93058)****5.**E. G. Gladyshev,*Periodically correlated random sequences*, Doklady Akad. Nauk SSSR**137**(1961), no. 5, 1026-1029; English transl. in Soviet Math. Dokl.**2**(1961), 385-388. MR**0126873 (23:A4167)****6.**U. Grenander,*A prediction problem in game theory*, Ark. Mat.**3**(1957), 371-379. MR**0090486 (19:822g)****7.**H. L. Hurd and A. Miamee,*Periodically Correlated Random Sequences: Spectral Theory and Practice*, John Wiley & Sons, 2007. MR**2348769 (2009e:62002)****8.**A. N. Kolmogorov,*Probability Theory and Mathematical Statistics*, Collection of problems, ``Nauka'', Moscow, 1986; English transl., Selected works, vol. II, Mathematics and its Applications (Soviet Series), vol. 26, Kluwer Academic Publishers Group, Dordrecht, 1992; translated from the Russian by G. Lindquist; translation edited by A. N. Shiryayev. MR**1153022 (92j:01071)****9.**A. Makagon,*Theoretical prediction of periodically correlated sequences*, Probab. Math. Statist.**19**(1999), 287-322. MR**1750905 (2001m:60093)****10.**A. Makagon,*Stationary sequences associated with a periodically correlated sequence*, Probab. Math. Statist.**31**(2011), 263-283. MR**2853678 (2012j:60091)****11.**M. P. Moklyachuk,*Estimates of stochastic processes from observations with noise*, Theory Stoch. Process.**3**(**19**) (1997), no. 3-4, 330-338.**12.**M. P. Moklyachuk,*Robust procedures in time series analysis*, Theory Stoch. Process.**6**(**22**) (2000), no. 3-4, 127-147.**13.**M. P. Moklyachuk,*Game theory and convex optimization methods in robust estimation problems*, Theory Stoch. Process.**7**(**23**) (2001), no. 1-2, 253-264.**14.**M. P. Moklyachuk,*Robust Estimates for Functionals of Stochastic Processes*, ``Kyivs'kyi Universytet'', Kyiv, 2008. (Ukrainian)**15.**M. P. Moklyachuk and O. Yu. Masyutka,*On the problem of filtration for stationary vector sequences*, Teor. Imovir. Matem. Statyst.**75**(2007), 95-104; English transl. in Theor. Probability Math. Statist.**75**(2007), 109-119. MR**2321185****16.**B. N. Pshenichnyĭ,*Necessary Conditions for an Extremum*, ``Nauka'', Moscow, 1982; English transl., Marcel Dekker, Inc., New York, 1971. MR**686452 (84c:49003)****17.**Yu. A. Rozanov,*Stationary Random Processes*, 2nd ed., ``Nauka'', Moscow, 1990. (Russian) MR**1090826 (92d:60046)****18.**N. Wiener,*Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications*, The M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass., 1966. MR**0031213 (11:118j)****19.**A. M. Yaglom,*Correlation Theory of Stationary and Related Random Functions. Vol. 1: Basic Results*, Springer Series in Statistics, Springer-Verlag, New York, etc., 1987. MR**893393 (89a:60105)****20.**A. M. Yaglom,*Correlation Theory of Stationary and Related Random Functions. Vol. 2: Supplementary Notes and References*, Springer Series in Statistics, Springer-Verlag, New York, etc., 1987. MR**915557 (89a:60106)**

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Additional Information

**I. I. Dubovets′ka**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine

Email:
idubovetska@gmail.com

**M. P. Moklyachuk**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine

Email:
mmp@univ.kiev.ua

DOI:
https://doi.org/10.1090/S0094-9000-2013-00888-6

Keywords:
Periodically correlated sequence,
robust estimate,
mean square error,
least favorable spectral density,
minimax spectral characteristic

Received by editor(s):
November 21, 2011

Published electronically:
August 20, 2013

Article copyright:
© Copyright 2013
American Mathematical Society