  In a recent paper of Eichelsbacher and Koenig (2008) the model of ordered
random walks has been considered. There it has been shown that, under certain
moment conditions, one can construct a k-dimensional random walk conditioned to
stay in a strict order at all times. Moreover, they have shown that the
rescaled random walk converges to the Dyson Brownian motion. In the present
paper we find the optimal moment assumptions for the construction of the
conditional random walk and generalise the limit theorem for this conditional
process.

Probability