An estimate of the mean square error of interpolation of stochastic processes. (Ukrainian, English) Zbl 1124.60035

Olenko, A. Ya.

Teor. Jmovirn. Mat. Stat. 72, 101-110 (2005); translation in Theory Probab. Math. Stat. 72, 113-123 (2006).

For stochastic processes of the weak Cramér class \(\xi(t)=\int_{\Lambda}f(t,\lambda)Z_{\xi}(d\lambda)\) with the correlation function \(B(t,s)=\int_{\Lambda}{\int_{\Lambda}}^{\ast} f(t,\lambda)\overline{ f(s,\mu)}F_{\xi}(d\lambda,d\mu)\) the author considers the approximation problem \[ \xi(t)\approx Y_{\mathcal J,\omega}(\xi,t)=\sum_{n\in\mathcal J}\sin(t\omega-n)\xi({n}/{\omega}) \] in the \(L_2(\Omega)\) sense. The optimal \(\varepsilon\) is found such that \(E| \xi(t)- Y_{\mathcal J,\omega}(\xi,t)| ^2<\varepsilon \| F_{\xi}\| (\Lambda,\Lambda)\). The rates of the mean square convergence of the error of interpolation for stochastic processes of the weak Cramér class and for processes generated by an orthogonal stochastic measure are estimated.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G12	General second-order stochastic processes
94A20	Sampling theory in information and communication theory
26D15	Inequalities for sums, series and integrals
30D15	Special classes of entire functions of one complex variable and growth estimates
41A05	Interpolation in approximation theory

Keywords:

Whittaker-Kotelnikov-Shannon interpolation; mean square error; Cramér class; stochastic measure