Wide sense stationary solutions of difference equations in a Banach space

Author:
M. F. Gorodnii

Translated by:
Oleg Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **74** (2006).

Journal:
Theor. Probability and Math. Statist. **74** (2007), 29-35

MSC (2000):
Primary 60G10, 39A10; Secondary 47A50

Published electronically:
June 25, 2007

MathSciNet review:
2336776

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A criterion is proved for the existence of a unique wide sense stationary solution of a linear difference equation with operator coefficients in a Banach space. The stability of this solution with respect to small perturbations of operator coefficients is proved.

**1.**M. F. Gorodnīĭ,*Solutions of a stochastic difference equation that are stationary and bounded in the mean*, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki**8**(2002), 12–16 (Ukrainian, with English summary). MR**2009511****2.**S. Bochner and R. S. Phillips,*Absolutely convergent Fourier expansions for non-commutative normed rings*, Ann. of Math. (2)**43**(1942), 409–418. MR**0007939****3.**M. F. Gorodniĭ,*Bounded and periodic solutions of a difference equation and its stochastic analogue in a Banach space*, Ukrain. Mat. Zh.**43**(1991), no. 1, 41–46 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J.**43**(1991), no. 1, 32–37. MR**1098269**, 10.1007/BF01066900**4.**V. M. Kruglov,*Dopolnitelnye glavy teorii veroyatnostei*, “Vyssh. Shkola”, Moscow, 1984 (Russian). MR**756812****5.**L. V. Kantorovich and G. P. Akilov,*Functional analysis*, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR**664597****6.**A. Ya. Dorogovtsev,*Stability of periodic solutions of operator equations with perturbation coefficients*, Exploring stochastic laws, VSP, Utrecht, 1995, pp. 111–119. MR**1713997****7.**A. Ya. Dorogovtsev,*Stability of bounded and stationary solutions of linear equations with respect to perturbations of operator coefficients*, Dokl. Akad. Nauk**345**(1995), no. 4, 448–450 (Russian). MR**1376444**

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Additional Information

**M. F. Gorodnii**

Affiliation:
Department of Mathematical Analysis, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine

Email:
gorodnii@yandex.ru

DOI:
http://dx.doi.org/10.1090/S0094-9000-07-00695-3

Keywords:
Difference equation,
operator coefficients,
wide sense stationary solutions

Received by editor(s):
March 18, 2005

Published electronically:
June 25, 2007

Article copyright:
© Copyright 2007
American Mathematical Society