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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A local limit theorem and loss of rotational symmetry of planar symmetric simple random walk


Author: Christian Beneš
Journal: Trans. Amer. Math. Soc. 371 (2019), 2553-2573
MSC (2010): Primary 60F05, 60G50
DOI: https://doi.org/10.1090/tran/7399
Published electronically: November 27, 2018
MathSciNet review: 3896089
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Abstract: We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of $ n$ and $ x$, where $ x$ is thought of as dependent on $ n$, in dimensions one and two, for $ P(S_n=x)$, the probability that symmetric simple random walk $ S$ started at the origin is at some point $ x$ at time $ n$, that are valid for all $ x$. We also show that the behavior of planar symmetric simple random walk differs radically from that of planar standard Brownian motion outside the disk of radius $ n^{3/4}$, where the random walk ceases to be approximately rotationally symmetric. Indeed, if $ n^{3/4}=o(\vert S_n\vert)$, $ S_n$ is more likely to be found along the coordinate axes. In this paper, we give a description of how the transition from approximate rotational symmetry to complete concentration of $ S$ along the coordinate axes occurs.


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Additional Information

Christian Beneš
Affiliation: Department of Mathematics, Brooklyn College, CUNY, Brooklyn, New York 11210
Email: CBenes@brooklyn.cuny.edu

DOI: https://doi.org/10.1090/tran/7399
Received by editor(s): December 30, 2016
Received by editor(s) in revised form: August 18, 2017, and September 6, 2017
Published electronically: November 27, 2018
Article copyright: © Copyright 2018 American Mathematical Society