Minimax interpolation of periodically correlated processes. (Q2850921)

The authors deal with the problem of mean square optimal estimation of the linear functional \(A_N\zeta=\int\limits_0^{(N+1)T} a(t)\zeta(t)dt\) depending on the unknown values of a periodically correlated stochastic process \(\zeta(t)\) with period \(T\) based on observations of the process \(\zeta(t)+\theta(t)\) at points of time \(t\in \mathbb R \backslash [0,(N+1)T] \), where \(\theta(t)\) is uncorrelated with \(\zeta(t)\), a periodically correlated process with period \(T\). Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional \(A_N\xi\) are proposed under the condition of spectral certainty, where the spectral densities of the processes \(\zeta(t)\) and \(\theta(t)\) are exactly known. The minimax (robust) method of estimation is used in the case where spectral densities are not known exactly but a set of admissible spectral densities is given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are proposed for some special sets of admissible densities. For more results and references, see [\textit{M. Moklyachuk} and \textit{O. Masyutka}, Minimax-robust estimation technique for stationary stochastic processes. Saarbrücken: LAP Lambert Academic Publishing (2012; Zbl 1289.62001); \textit{M. P. Moklyachuk}, Robust estimates for functionals of stochastic processes. Kyïv: Vydavnycho-Poligrafichnyĭ\ Tsentr, Kyïvskyĭ\ Universytet (2008; Zbl 1249.62007)].