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# Realizations of stationary stochastic processes: applications of passive system theory

### Abstract

In the paper, we investigate realizations of a $p$-dimensionalregular weak stationary discrete time stochastic process $y(t)$ asthe output data of a passive linear bi-stable discrete timedynamical system. The state $x(t)$ is assumed to tend to zero as ttends to $-\infty$, and the input data is the $m$-dimensional whitenoise. The results are based on author's development of the Darlington method for passive impedance systems with losses of thescattering channels. Here we establish that considering realizationfor a discrete time process is possible, if the spectral density $\rho(e^{i\mu})$ of the process is a nontangential boundary value ofa matrix valued meromorphic function $\rho(z)$ of rank $m$ withbounded Nevanlinna characteristic in the open unitdisk. A parameterization of all such realizations is given and minimal,optimal minimal, and *-optimal minimal realizations areobtained. The last two coincide with those which are obtained by Kalman filters. This is a further development of the Lindquist-Picci realization theory.

### Article Information

 Title Realizations of stationary stochastic processes: applications of passive system theory Source Methods Funct. Anal. Topology, Vol. 18 (2012), no. 4, 305-331 MathSciNet MR3058459 zbMATH 1289.60065 Copyright The Author(s) 2012 (CC BY-SA)

### Authors Information

D. Z. Arov
South Ukrainian National K. D. Ushyns'ky Pedagogical University, 26 Staroportofrankivs'ka, Odessa, 65020, Ukraine

N. A. Rozhenko
University Brunei Darussalam, FOS UBD, Jln Tungku Link, Gadong, BE1410, Brunei

### Citation Example

D. Z. Arov and N. A. Rozhenko, Realizations of stationary stochastic processes: applications of passive system theory, Methods Funct. Anal. Topology 18 (2012), no. 4, 305-331.

### BibTex

@article {MFAT658,
AUTHOR = {Arov, D. Z. and Rozhenko, N. A.},
TITLE = {Realizations of stationary stochastic processes: applications of passive system theory},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {18},
YEAR = {2012},
NUMBER = {4},
PAGES = {305-331},
ISSN = {1029-3531},
MRNUMBER = {MR3058459},
ZBLNUMBER = {1289.60065},
URL = {http://mfat.imath.kiev.ua/article/?id=658},
}