The two-sided exit problem for a random walk on ℤ with infinite variance I

2019/08/01 by Kôhei Uchiyama, Uchiyama, Kohei
Decision Sciences · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Primary 60G50 #Probability (math.PR) #Probability and Risk Models #Secondary 60J45 #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1908.00303

openalex publication_date 2019/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S=(S_n) be an oscillatory random walk on the integer lattice ℤ with i.i.d. increments. Let V_{\rm d}(x) be the renewal function of the strictly descending ladder height process for S. We obtain several sufficient conditions -- given in terms of the distribution function of the increment S₁-S₀ -- so that as R→∞ (*) P [ S leaves [0,R] on its upper side | S₀=x] ∼ V_{\rm d}(x)/V_{\rm d}(R) uniformly for 0≤ x≤ R. When S is attracted to a stable process of index 00], the sufficient condition obtained are also necessary for (*) and fulfilled if and only if (α\vee 1)ρ=1, and some asymptotic estimates of the probability on the left side of (*) are given in case (α\vee 1)ρ≠ 1.

Citations

• Estimates of Potential functions of random walks on Z with zero mean and infinite variance and their applications

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