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Limit behavior of symmetric random walks with a membrane


Authors: A. Yu. Pilipenko and Yu. E. Pryhod’ko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 85 (2011).
Journal: Theor. Probability and Math. Statist. 85 (2012), 93-105
MSC (2010): Primary 60F17, 60J10
Published electronically: January 14, 2013
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Abstract: Let $ \{X(k), k\in \mathbb{Z}_+\}$ be a random walk in $ \mathbb{Z}$. Assume that its transition probabilities coincide with those of a symmetric random walk with unit steps throughout except for a fixed neighborhood of zero. The weak convergence of the sequence of normalized walks $ \{X_n(k) = n^{-1/2} X(nk),k\geqslant 0\}_{n\geqslant 1}$ is proved. The main result generalizes a Harrison and Shepp theorem on the weak convergence to a skew Brownian motion in the case where the symmetricity of the random walk fails at a single point. All possible limits for the corresponding random walks are described.


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Additional Information

A. Yu. Pilipenko
Affiliation: Institute of Mathematics of National Academy of Science of Ukraine, Tereshchenkivs’ka Street 3, 01601, Kyiv, Ukraine
Email: apilip@imath.kiev.ua

Yu. E. Pryhod’ko
Affiliation: National Technical University of Ukraine “KPI”, Department of Mathematical Analysis and Probability Theory, Peremogy Avenue 37, 03056, Kyiv, Ukraine
Email: npuxodbko@gmail.com

DOI: http://dx.doi.org/10.1090/S0094-9000-2013-00877-1
Keywords: Random walks, skew Brownian motion, diffusion with a membrane
Received by editor(s): September 3, 2010
Published electronically: January 14, 2013
Additional Notes: Partially supported by a grant of State Foundation for Fundamental Researches of Ukraine and Russian Foundation for Fundamental Researches (grant # $Φ$40.1/023)
Article copyright: © Copyright 2013 American Mathematical Society