  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.
