On a structure of a conditioned random walk on the integers with bounded local times

We consider a sample path of a random walk on the integers with bounded local times, conditioned on the event that it hits a high level. Under an auxiliary assumption, we obtain representations for its distribution in terms of the corresponding limiting sequence. Then we prove limiting results as the high level grows. In particular, we generalize results for a simple symmetric random walk obtained earlier by Benjamini and Berectycki (2010).