Random walks with square-root boundaries: the case of exact boundaries g(t)=c√(t+b)-a

2025/01/08 by Denis Denisov, Denisov, Denis, A. I. Sakhanenko +5
Mathematics · #60G50 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2501.04554

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

Abstract

Let S(n) be a real valued random walk with i.i.d. increments which have zero mean and finite variance. We are interested in the asymptotic properties of the stopping time T(g):=inf\n≥1: S(n)≤ g(n)\, where g(t) is a boundary function. In the present paper we deal with the parametric family of boundaries \g_a,b(t)=c√(t+b)-a, b≥0, a>c√(b)\. First, assuming that sufficiently many moments of increments of the walk are finite, we construct a positive space-time harmonic function W(a,b). Then we show that there exist p(c)>0 and a constant \varkappa(c) such that P(T_g_a,b>n)∼ \varkappa(c)\fracW(a,b)n^p(c)/2 as n→∞.

Related

• Green's Functions for Random Walks on ZN
• Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem
• Conditioned local limit theorems for random walks on the real line
• First-passage times over moving boundaries for asymptotically stable walks
• Local limit theorems for conditioned random walks by the heat kernel approximation