Electronic Journal of Probability

Ordered Random Walks

Peter Eichelsbacher and Wolfgang König

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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.

Article information

Electron. J. Probab., Volume 13 (2008), paper no. 46, 1307-1336.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F17: Functional limit theorems; invariance principles

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

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Eichelsbacher, Peter; König, Wolfgang. Ordered Random Walks. Electron. J. Probab. 13 (2008), paper no. 46, 1307--1336. doi:10.1214/EJP.v13-539. https://projecteuclid.org/euclid.ejp/1464819119

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