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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 46 open journal systems 


Ordered Random Walks

Peter Eichelsbacher, University of Bochum
Wolfgang König, University of Leipzig


Abstract
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.


Full text: PDF

Pages: 1307-1336

Published on: August 14, 2008
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Electronic Journal of Probability. ISSN: 1083-6489