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Wide sense stationary solutions of difference equations in a Banach space
Author(s):
M.
F.
Gorodnii
Translated by:
Oleg Klesov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 74
(2006).
Journal:
Theor. Probability and Math. Statist.
No. 74
(2007),
29-35.
MSC (2000):
Primary 60G10, 39A10;
Secondary 47A50
Posted:
June 25, 2007
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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.
References:
-
- 1.
- M. F. Gorodni
 , 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. MR 2009511 (2004g:60099) - 2.
- S. Bochner and R. S. Phillips, Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. Math. 43 (1942), no. 3, 409-418. MR 0007939 (4:218g)
- 3.
- M. F. Gorodni
 , 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; English transl. in Ukrainian Math. J. 43 (1991), no. 1, 32-37. MR 1098269 (92d:60068) - 4.
- V. M. Kruglov, Supplementary Chapters on Probability Theory, ``Vysshaya Shkola'', Moscow, 1984. (Russian) MR 756812 (86d:60009)
- 5.
- L. V. Kantorovich and G. P. Akilov, Functional Analysis, ``Nauka'', Moscow, 1984; English transl., Pergamon Press, Oxford-Elmsford, N.Y., 1982. MR 664597 (83h:46002)
- 6.
- A. Ya. Dorogovtsev, Stability of periodic solutions of operator equations with perturbation coefficients, Exploring Stochastic Laws (A. V. Skorokhod and Yu. V. Borovskikh, eds.), VSP, Utrecht, The Netherlands, 1995, pp. 111-119. MR 1713997 (2001g:60163)
- 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. MR 1376444 (96k:47123)
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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:
10.1090/S0094-9000-07-00695-3
PII:
S 0094-9000(07)00695-3
Keywords:
Difference equation,
operator coefficients,
wide sense stationary solutions
Received by editor(s):
18/MAR/2005
Posted:
June 25, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
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