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2008 Ordered Random Walks
Peter Eichelsbacher, Wolfgang König
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Electron. J. Probab. 13: 1307-1336 (2008). DOI: 10.1214/EJP.v13-539


We construct the conditional version of $k$ independent and identically distributed random walks on $R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.


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Peter Eichelsbacher. Wolfgang König. "Ordered Random Walks." Electron. J. Probab. 13 1307 - 1336, 2008.


Accepted: 14 August 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1189.60092
MathSciNet: MR2430709
Digital Object Identifier: 10.1214/EJP.v13-539

Primary: 60G50
Secondary: 60F17

Keywords: Doob h-transform , Dyson's Brownian motions , fluctuation theory , non-colliding random walks , non-intersecting random processes , Vandermonde determinant


Vol.13 • 2008
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