Abstract
For a process X(t)=Σ M j=1 g j (t)ξ j (), where gj(t) are nonrandom given functions, \((\xi _j (t),j = \overline {1,M} )\) is a stationary vector-valued Gaussian process, Eξk(t) = 0, and Eξk(0) Eξl(τ) = r kl(τ), we construct an estimate \(\hat r_{kl} (\tau ,T)\) for the functions r kl(τ) on the basis of observations X(t), t ∈ [0, T]. We establish conditions for the asymptotic normality of \(\sqrt T (\hat r_{kl} (\tau ,T) - r_{kl} (\tau ))\) as T → ∞. We consider the problem of the optimal choice of parameters of the estimate \(\hat r_{kl} \) depending on observations.
Similar content being viewed by others
References
A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields [in Russian], Vyshcha Shkola, Kiev (1986). English translation: Kluwer AP, Dordrecht (1989).
V. V. Buldygin, “On the asymptotic properties of an empirical correlogram of a Gaussian process,” Dokl. Akad. Nauk Ukr., No. 11, 33–38 (1994).
V. V. Buldygin and O. O. Dem’yanenko, “Point properties of estimates of a joint correlation function of Gaussian fields,” in: Stochastic Equations and Limit Theorems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1995), pp. 23–35.
V. N. Vapnik, “Inductive principles of a search for empirical regularities,” in: Recognition-Classification-Prediction [in Russian], Issue 1 (1989), pp. 17–81.
A. B. Bakushinskii and A. V. Goncharskii, Ill-Posed Problems. Numerical Methods and Applications [in Russian], Moscow University, Moscow (1989).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 937–947, July, 1998.
Rights and permissions
About this article
Cite this article
Maiboroda, R.E. Asymptotic normality and efficiency of a weighted correlogram. Ukr Math J 50, 1067–1079 (1998). https://doi.org/10.1007/BF02528835
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02528835