A Divergent Random Walk on Stairs
Yufan Li, Jeffery Rosenthal

TL;DR
This paper studies a state-dependent random walk on natural numbers influenced by a sequence _n, showing that for any probability less than 1, a sequence exists making the walk diverge with at least that probability, but the case for probability 1 remains open.
Contribution
The paper proves that for any <<1, there exists a sequence _n making the walk diverge with probability at least , advancing understanding of the walk's divergence behavior.
Findings
Constructed sequences _n ensuring divergence with probability <1
Extended previous results by achieving divergence probability arbitrarily close to 1
The case for divergence probability exactly 1 remains unresolved.
Abstract
We consider a state-dependent, time-dependent, discrete random walks defined on natural numbers (bent to a "stair" in ) where the random walk depends on input of a positive deterministic sequence . This walk has the peculiar property that if we set to be for all , it converges to a stationary distribution ; but if is uniformly bounded (over all ) by any upper bound , this walk diverges to infinity with probability 1. It is thus interesting to consider the intermediate case where for all but eventually tends to . (Latuszynski et al., 2013) first defined this walk and conjectured that a particular choice of sequence exists such that (i) and, (ii) . They managed to construct a sequence…
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Taxonomy
TopicsStochastic processes and statistical mechanics · Markov Chains and Monte Carlo Methods · Limits and Structures in Graph Theory
