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An invariance principle and a large deviation principle for the biased random walk on ${\mathbb{Z}}^{\lowercase{\textbf{\textit{d}}}}$

Part of: Markov processes Limit theorems Graph theory

Published online by Cambridge University Press:  04 May 2020

Yuelin Liu ,
Vladas Sidoravicius ,
Longmin Wang  and
Kainan Xiang
Yuelin Liu*
Affiliation:
Nankai University
Vladas Sidoravicius*
Affiliation:
CIMS; NYU-Shanghai
Longmin Wang*
Affiliation:
Nankai University
Kainan Xiang*
Affiliation:
Xiangtan University
*
*Postal address: School of Mathematical Sciences, LPMC, Nankai University, Tianjin, 300071, China.
***Postal address: Courant Institute of Mathematical Sciences, New York, NY10012, USA; NYU-ECNU Institute of Mathematical Sciences, NYU-Shanghai, Shanghai, 200122, China.
*Postal address: School of Mathematical Sciences, LPMC, Nankai University, Tianjin, 300071, China.
****Postal address: School of Mathematics and Computational Science, Xiangtan University, Xiangtan City, 411105, China.
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Abstract

We establish an invariance principle and a large deviation principle for a biased random walk ${\text{RW}}_\lambda$ with $\lambda\in [0,1)$ on $\mathbb{Z}^d$ . The scaling limit in the invariance principle is not a d-dimensional Brownian motion. For the large deviation principle, its rate function is different from that of a drifted random walk, as may be expected, though the reflected biased random walk evolves like the drifted random walk in the interior of the first quadrant and almost surely visits coordinate planes finitely many times.

Keywords

Biased random walkinvariance principlelarge deviation principle

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C81: Random walks on graphs
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 295 - 313
Copyright
© Applied Probability Trust 2020

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