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

TL;DR

This work analyzes a one-dimensional random walk killed at crossing a square-root boundary $g_{a,b}(t)=c\sqrt{t+b}-a$, establishing sharp one-sided tail asymptotics for the crossing time. The authors develop a space-time potential theory framework, constructing a positive harmonic function $W(a,b)$ from Brownian-motivated functions $V_p$ and linking it to the discrete walk via a Doob $h$-transform. The main result shows $\mathbb{P}(T_{a,b}>n)\sim \kappa(c)\,W(a,b)/n^{p(c)/2}$, where $p(c)$ is the smallest positive root of $\psi_p(c)$ and $\kappa(c)$ is Uchiyama’s constant, under suitable moment conditions. By combining martingale methods, local probabilistic estimates, and Brownian comparison, the paper extends the square-root boundary analysis to exact one-sided boundaries, setting the stage for broader boundary classes in future work.

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 $n\to\infty$.

Theorems & Definitions (49)

• Theorem 1
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• Lemma 6
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• Lemma 7
• proof