Convergence of a sequence of Markov chains to a diffusion type process

Authors:
G. L. Kulinich and A. V. Yershov

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **78** (2008).

Journal:
Theor. Probability and Math. Statist. **78** (2009), 115-131

MSC (2000):
Primary 60H10; Secondary 34G10, 47A50, 47D06

Published electronically:
August 4, 2009

MathSciNet review:
2446853

Full-text PDF Free Access

Abstract | References | Similar Articles | 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.

**1.**I. I. Gikhman and A. V. Skorokhod,*Introduction to the theory of random processes*, Translated from the Russian by Scripta Technica, Inc, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR**0247660****2.**Ĭ. Ī. Gīhman and A. V. Skorohod,*Stochastic differential equations*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. MR**0346904****3.**N. V. Krylov,*Upravlyaaemye protsessy diffuzionnogo tipa*, “Nauka”, Moscow, 1977 (Russian). Teoriya Veroyatnostei i Matematicheskaya Statistika. [Monographs in Probability and Mathematical Statistics]. MR**508417**

N. V. Krylov,*Controlled diffusion processes*, Applications of Mathematics, vol. 14, Springer-Verlag, New York-Berlin, 1980. Translated from the Russian by A. B. Aries. MR**601776****4.**G. L. Kulīnīč,*Certain limit theorems for a sequence of Markov chains*, Teor. Verojatnost. i Mat. Statist.**Vyp. 12**(1975), 77–89, 172 (Russian, with English summary). MR**0397835****5.**Grigorii L. Kulinich and Eugenii P. Kaskun,*On the asymptotic behavior of solutions of one-dimensional Ito’s stochastic differential equations with singularity points*, Proceedings of the Donetsk Colloquium on Probability Theory and Mathematical Statistics (1998), 1998, pp. 189–197. MR**2026628****6.**G. L. Kulīnīch,*Necessary and sufficient conditions for the convergence of solutions of one-dimensional stochastic diffusion equations with irregular dependence of the coeffici*, Teor. Veroyatnost. i Primenen.**27**(1982), no. 4, 795–802 (Russian, with English summary). MR**681473****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*, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0185620**

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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:
https://doi.org/10.1090/S0094-9000-09-00766-2

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):
May 7, 2007

Published electronically:
August 4, 2009

Article copyright:
© Copyright 2009
American Mathematical Society