Duals of random vectors and processes with applications to prediction problems with missing values

摘要

Important results in prediction theory dealing with missing values have been obtained traditionally using difficult techniques based on duality in Hilbert spaces of analytic functions [Nakazi, T., 1984. Two problems in prediction theory. Studia Math. 78, 7-14; Miamee, A.G., Pourahmadi, M., 1988. Best approximations in L-p(d mu) and prediction problems of Szego, Kolmogorov, Yaglom, and Nakazi. J. London Math. Soc. 38, 133-145]. We obtain and unify these results using a simple finite-dimensional duality lemma which is essentially an abstraction of a regression property of a multivariate normal random vector (Rao, 1973) or its inverse covariance matrix. The approach reveals the roles of duality and biorthogonality of random vectors in dealing with infinite-dimensional and difficult prediction problems. A novelty of this approach is its reliance on the explicit representation of the prediction error in terms of the data rather than the predictor itself as in the traditional techniques. In particular, we find a new and explicit formula for the dual of the semi-finite process {X-t; t <= n} for a fixed n, which does not seem to be possible using the existing techniques.