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Theory of Probability and Mathematical Statistics

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Existence of a limit distribution of a solution of a linear inhomogeneous stochastic differential equation

Author: D. O. Ivanenko
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 49-60
MSC (2000): Primary 60F05; Secondary 60J75
Published electronically: August 4, 2009
MathSciNet review: 2446848
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Abstract | References | Similar Articles | Additional Information

Abstract: We find conditions for the existence of a limit distribution (as $ t\rightarrow\infty$) of a vector process $ \xi$ defined in $ \mathbb{R}_+$ and determined by an inhomogeneous stochastic differential equation $ \xi(t)=\xi(0)-\xi\circ\alpha+f\ast\nu+g\ast\mu$, where $ \alpha$ is a nonrandom continuous increasing function, $ \nu$ and $ \mu$ are independent Poisson and centered Poisson measures, respectively.

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  • 1. I. I. Gikhman and A. V. Skorokhod, \cyr Stokhasticheskie differentsial′nye uravneniya i ikh prilozheniya, “Naukova Dumka”, Kiev, 1982 (Russian). MR 678374
  • 2. R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664
  • 3. O. K. Zakusilo, The classes of limit distributions in a certain summation scheme, Teor. Verojatnost. i Mat. Statist. Vyp. 12 (1975), 44–48, 172 (Russian, with English summary). MR 0397833
  • 4. O. K. Zakusilo, Some properties of random vectors of the form ∑^{∞}ᵢ₌₀𝐴ⁱ𝜉ᵢ, Teor. Verojatnost. i Mat. Statist. Vyp. 13 (1975), 59–62, 162 (Russian, with English summary). MR 0415734
  • 5. V. V. Anisimov, O. K. Zakusilo, and V. S. Donchenko, Elements of Queueing Theory and Asymptotic Analysis of Systems, Vyshcha shkola, Kiev, 1987. (Russian)
  • 6. Hiroki Masuda, On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process, Bernoulli 10 (2004), no. 1, 97–120. MR 2044595,
  • 7. Hiroki Masuda, Ergodicity and exponential 𝛽-mixing bounds for multidimensional diffusions with jumps, Stochastic Process. Appl. 117 (2007), no. 1, 35–56. MR 2287102,
  • 8. Ken-iti Sato and Makoto Yamazato, Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type, Stochastic Process. Appl. 17 (1984), no. 1, 73–100. MR 738769,
  • 9. A. N. Kolmogorov, \cyr Osnovnye ponyatiya teorii veroyatnosteĭ, 2nd ed., Izdat. “Nauka”, Moscow, 1974 (Russian). Probability Theory and Mathematical Statistics Series, Vol. 16. MR 0353394
    A. N. Kolmogorov, Foundations of the theory of probability, Chelsea Publishing Co., New York, 1956. Translation edited by Nathan Morrison, with an added bibliography by A. T. Bharucha-Reid. MR 0079843

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Additional Information

D. O. Ivanenko
Affiliation: Department of Mathematics and Theoretical Radiophysics, Faculty for Radiophysics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Keywords: Limit distribution, Poisson measure, It\^o's formula, Tauberian theorem
Received by editor(s): July 3, 2007
Published electronically: August 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society