2018/08/30 by Yufan Li, Li, Yufan, Jeffery Rosenthal +1
Mathematics · #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1808.10121
openalex publication_date 2018/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a state-dependent, time-dependent, discrete random walks Xt^\an\ defined on natural numbers ℕ (bent to a "stair" in ℕ2) where the random walk depends on input of a positive deterministic sequence \an\. This walk has the peculiar property that if we set an to be +∞ for all n, it converges to a stationary distribution π(⋅); but if an is uniformly bounded (over all n) by any upper bound a ∈ (0,∞), this walk diverges to infinity with probability 1. It is thus interesting to consider the intermediate case where an0, which is weaker than (ii). In this paper, we obtain a stronger result: for any σ<1, there exists a choice of \an\ so that P(Xt→ ∞)≥ σ. Our result does not apply when σ=1, the original conjecture remains open. We record our method here for technical interests.