Asymptotic normality and efficiency of a weighted correlogram

Abstract

For a process X(t)=Σ _j=1 ^M g _j (t)ξ _j (), where g_j(t) are nonrandom given functions, $(\xi _j (t),j = \overline {1,M} )$ is a stationary vector-valued Gaussian process, Eξ_k(t) = 0, and Eξ_k(0) Eξ_l(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.