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Convergence of a sequence of Markov chains to a diffusion type process
Author(s):
G.
L.
Kulinich;
A.
V.
Yershov
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 78
(2008).
Journal:
Theor. Probability and Math. Statist.
No. 78
(2009),
115-131.
MSC (2000):
Primary 60H10;
Secondary 34G10, 47A50, 47D06
Posted:
August 4, 2009
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Additional information
Abstract:
A random polygonal line constructed from a sequence of series of homogeneous Markov chains is considered under rather nonregular dependence of their local characteristics on a series number. Sufficient conditions are obtained for the weak convergence of a random polygonal line to a diffusion type process. The conditions are expressed explicitly in terms of local characteristics of the Markov chains.
References:
-
- 1.
- I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, Nauka, Moscow, 1965; English transl., W. B. Saunders Co., Philadelphia, 1969. MR 0247660 (40:923)
- 2.
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Naukova Dumka, Kiev, 1968; English transl., Springer-Verlag, Berlin-Heidelberg-New York, 1972. MR 0346904 (49:11625)
- 3.
- N. V. Krylov, Controlled Diffusion Processes, Nauka, Moscow, 1977; English transl., Springer-Verlag, Berlin-New York, 1980. MR 508417 (80f:60046); MR 0601776 (82a:60062)
- 4.
- G. L. Kulinich, Some limit theorems for a sequence of Markov chains, Teor. Veroyatnost. Matem. Statist. 12 (1975), 77-89; English transl. in Theory Probab. Math. Statist. 12 (1976), 79-92.MR 0397835 (53:1691)
- 5.
- G. L. Kulinich and E. P. Kaskun, On the asymptotic behavior of solutions of one-dimensional Ito's stochastic differential equations with singularity points, Theory Stoch. Process. 4(20) (1998), no. 1-2, 189-197. MR 2026628 (2004j:60122)
- 6.
- G. L. Kulinich, Necessary and sufficient conditions for the convergence of solutions of one-dimensional stochastic differential equations with irregular dependence of the coefficients on a parameter, Teor. Veroyatnost. Primenen. 27 (1982), no. 4, 795-801; English transl. in Theory Probab. Appl. 27 (1982), no. 4, 856-862. MR 681473 (84g:60093)
- 7.
- G. L. Kulinich, Asymptotic Analysis of Unstable Solutions of One-Dimensional Stochastic Differential Equations, Kyiv University, Kyiv, 2003. (Ukrainian)
- 8.
- A. V. Skorokhod, Studies in the Theory of Random Processes, Kiev University, Kiev, 1961; English transl., Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0185620 (32:3082b)
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Additional Information:
G.
L.
Kulinich
Affiliation:
Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
a_yershov@univ.kiev.ua
A.
V.
Yershov
Affiliation:
Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
DOI:
10.1090/S0094-9000-09-00766-2
PII:
S 0094-9000(09)00766-2
Keywords:
A sequence of series of Markov chains,
nonregular dependence of local characteristics of Markov chains on the number of a series,
a random polygonal line,
weak convergence,
stochastic differential equation,
diffusion type processes
Received by editor(s):
7/MAY/2007
Posted:
August 4, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
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