On the Mean Convergence of the Best Linear Interpolator of Multivariate Stationary Stochastic Processes

Abstract

It is shown that a necessary and sufficient condition, for the existence of a mean-convergent series for the linear interpolator of a $q$-variate stationary stochastic process $\{X_n\}$ with density matrix $W$, is that the Fourier series of the isomorph of the linear interpolator should converge in the norm of $L^2(W)$, and this happens if the past and future of the process are at positive angle. This provides a positive answer to a question of H. Salehi (1979) concerning the square summability of the inverse of $W$ and improves upon the work of Rozanov (1960) and Salehi (1979).