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IMA Journal of Mathematical Control and Information 1989 6(3):347-358; doi:10.1093/imamci/6.3.347
© 1989 by Institute of Mathematics and its Applications
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A New Approach to the Linear Prediction Problem for Infinite Stationary Processes in Terms of Riemann-Hilbert Transformation Theory

YOSHIMASA NAKAMURA

Department of Mathematics, Faculty of Education, Gifu University Gifu 501-11, Japan

A group-theoretic structure of the space of linear predictors for multivariate infinite stationary processes is discussed by applying tie Riemann-Hilbert (RH) transformation theory developed in the study of nonlinear integrable equations. It is proved that RH transformations induce fractional transformations from the set of old linear predictors to a new one. Infinitesimal RH transformations are found to form the Kac-Moody algebra without centre acting on the space of linear predictors. It is shown that the RH transformations define a dynamical system on an infinite-dimensional Grassmann manifold which is governed by a matrix Riccati equation. Parametric linear predictors are obtained by exponentiating infinitesimal RH transformations. These are regarded as matrix representations of Banach Lie groups acting on the space of linear predictors. Another class of RH transformations is also considered for a backward linear prediction problem.


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