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 proposed by Eichelsbacher and Koenig, and generalise the limit theorem for this conditional process.
"Conditional Limit Theorems for Ordered Random Walks." Electron. J. Probab. 15 292 - 322, 2010. https://doi.org/10.1214/EJP.v15-752