On representations and simulation of conditioned random walks on integer lattices

Abstract

We consider a random walk on a multidimensional integer lattice with random bounds on local times. We introduce a family of auxiliary ``accompanying'' processes that have regenerative structures and play key roles in our analysis. We obtain a number of representations for the distribution of the random walk in terms of similar distributions of the ``accompanying'' processes. Based on that, we obtain representations for the conditional distribution of the random walk, conditioned on the event that it hits a high level before its death. Under more restrictive assumptions a representation of such type has been obtained earlier by the same authors in a recent paper published in the Springer series on Progress in Probability, 77 (2021), where a certain ``limiting'' process was used in place of ``accompanying'' processes of the present paper.