Conditional limit theorems for ordered random walks

TL;DR

This paper refines the understanding of ordered random walks by establishing optimal moment conditions for their construction and extending the limit theorem to a broader class of these processes, connecting them to Dyson Brownian motion.

Contribution

It identifies the minimal moment assumptions needed for constructing ordered random walks and generalizes the convergence results to the associated conditional processes.

Findings

Optimal moment conditions for ordered random walk construction

Generalization of the limit theorem for conditional processes

Connection to Dyson Brownian motion

Abstract

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.