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2012 Two-sided random walks conditioned to have no intersections
Daisuke Shiraishi
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Electron. J. Probab. 17: 1-24 (2012). DOI: 10.1214/EJP.v17-1742

Abstract

Let $S^{1},S^{2}$ be independent simple random walks in $\mathbb{Z}^{d}$ ($d=2,3$) started at the origin. We construct two-sided random walk paths conditioned that $S^{1}[0,\infty ) \cap S^{2}[1, \infty ) = \emptyset$ by showing the existence of the following limit:\begin{equation*}\lim _{n \rightarrow \infty } P ( \cdot | S^{1}[0, \tau ^{1} ( n) ] \cap S^{2}[1, \tau ^{2}(n) ] = \emptyset ),\end{equation*}where $\tau^{i}(n) = \inf \{ k \ge 0 : |S^{i} (k) | \ge n \}$. Moreover, we give upper bounds of the rate of the convergence. These are discrete analogues of results for Brownian motion obtained by Lawler.

Citation

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Daisuke Shiraishi. "Two-sided random walks conditioned to have no intersections." Electron. J. Probab. 17 1 - 24, 2012. https://doi.org/10.1214/EJP.v17-1742

Information

Accepted: 28 February 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1244.05209
MathSciNet: MR2900459
Digital Object Identifier: 10.1214/EJP.v17-1742

Subjects:
Primary: 05C81

JOURNAL ARTICLE
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