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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a random walk in . 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 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