On stochastic equation describing the one-sided moving average process and minimax estimation problem

On stochastic equation describing the one-sided moving average process and minimax estimation... Random Oper. & Stock. Eqs., Vol. 1, No. 4, pp. 329-343 (1993) O VSP 1993 M. P. MOKLYACHUK Department of Mechanics and Mathematics, Kyjiv University, 252017 Kyjiv, Ukraine Received for ROSE 15 June 1991 Abstract--The problem of optimal linear estimation of the transformation = f ·/o of a stationary stochastic process £(t) with values in a Hubert space from observations f(t) as t ^ 0 is considered. The minimax spectral characteristics of the optimal estimate of the transformation and the least favourable spectral densities for the various classes of densities are found. Denote by X a separable Hubert space with inner product (x,y) and an orthonormal basis {ek: k = 1,2,...}. A stochastic process £(t) with values in X is stationary, if its components £* = (f(£),ejt) are square-mean continuous and satisfy the conditions from [1, 2]: Efc(i) = 0, E|K(t)|& = V E|fc(t)|a < oo, The correlation function B(t) of such a process is a weakly continuous operator- valued function on X. The correlation operator B = JE?(0) is nuclear: The stochastic process £(t) has the spectral density /() if the correlation function B(i) can be represented in the form ej) = 1- e itA (/(A)e fc http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Random Operators and Stochastic Equations de Gruyter

On stochastic equation describing the one-sided moving average process and minimax estimation problem

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0926-6364
eISSN
1569-397X
DOI
10.1515/rose.1993.1.4.329
Publisher site
See Article on Publisher Site

Abstract

Random Oper. & Stock. Eqs., Vol. 1, No. 4, pp. 329-343 (1993) O VSP 1993 M. P. MOKLYACHUK Department of Mechanics and Mathematics, Kyjiv University, 252017 Kyjiv, Ukraine Received for ROSE 15 June 1991 Abstract--The problem of optimal linear estimation of the transformation = f ·/o of a stationary stochastic process £(t) with values in a Hubert space from observations f(t) as t ^ 0 is considered. The minimax spectral characteristics of the optimal estimate of the transformation and the least favourable spectral densities for the various classes of densities are found. Denote by X a separable Hubert space with inner product (x,y) and an orthonormal basis {ek: k = 1,2,...}. A stochastic process £(t) with values in X is stationary, if its components £* = (f(£),ejt) are square-mean continuous and satisfy the conditions from [1, 2]: Efc(i) = 0, E|K(t)|& = V E|fc(t)|a < oo, The correlation function B(t) of such a process is a weakly continuous operator- valued function on X. The correlation operator B = JE?(0) is nuclear: The stochastic process £(t) has the spectral density /() if the correlation function B(i) can be represented in the form ej) = 1- e itA (/(A)e fc

Journal

Random Operators and Stochastic Equationsde Gruyter

Published: Jan 1, 1993

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