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[We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary “core” process that has a regenerative structure and plays a key role in our analysis. We obtain a number of representations for the distribution of the random walk in terms of the similar distribution of the “core” process. Based on that, we prove a number of limiting results by letting the high level to tend to infinity. In particular, we generalise results for a simple symmetric one-dimensional random walk obtained earlier in the paper by Benjamini and Berestycki (J Eur Math Soc 12(4):819–854, 2010).]
Published: Nov 4, 2020
Keywords: Conditioned random walk; Bounded local times; Regenerative sequence; Potential regeneration; Separating levels; Skip-free distributions; 60K15; 60K37; 60F99; 60G99
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