Random walks with square-root boundaries: the case of exact boundaries $g(t)=c\sqrt{t+b}-a$

TL;DR

This paper analyzes the asymptotic behavior of the stopping time for a random walk crossing a specific square-root boundary, providing explicit probability decay rates and harmonic functions under certain moment conditions.

Contribution

It introduces a new harmonic function for the boundary crossing problem and derives asymptotic probabilities for the stopping time in the case of square-root boundaries.

Findings

Existence of a positive harmonic function W(a,b) for the boundary crossing problem.

Asymptotic probability of the stopping time exceeding n behaves like a constant times n^{-p(c)/2}.

Explicit decay rate and harmonic function for the boundary crossing problem.

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 $T(g):=\inf\{n\ge1: S(n)\le g(n)\}$, where $g(t)$ is a boundary function. In the present paper we deal with the parametric family of boundaries $\{g_{a,b}(t)=c\sqrt{t+b}-a, b\ge0, a>c\sqrt{b}\}$. 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 $\mathbf{P}(T_{g_{a,b}}>n)\sim \varkappa(c)\frac{W(a,b)}{n^{p(c)/2}}$ as $n\to\infty$.