2006/10/27 by Eichelsbacher, Peter, Konig, Wolfgang · 4 citations
#60F17 #60G50 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.math/0610850
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.