On a relationship between the inverse of a stationary covariance matrix and the linear interpolator

Abstract

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices ofxB(k) = [x′–1,x′–2, · ··,x′–k]′,xF(k) = [x′1,x′2· ··x′k] andx= [·· ·x′–2,x′–1,x′0,x′1,x′2· ··]′ respectively, wherek≧ 1, is finite or infinite. Also letφm,n(j) and δm,n(u) be the coefficients ofxt+jandxt–urespectively in the linear least-squares interpolator ofxtfromxt+ 1, · ··,xt+m;xt− 1, · ··,xt–n, wherem, n≧ 0, 0 ≦j≦m, 0 ≦u≦nare integers,zt(m, n) denote the interpolation error and τ2(m, n) =E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1and Γ–1is given by relating these components to theφm,n(j)δm,n(u) andτ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–land the interpolation parameters when these have been obtained from a realization of lengthTof {xt}. Some of the applications of the results are considered.