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

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 $T(g):=\inf \{n\ge 1:S(n)\le g(n)\}$, where $g(t)$ is a boundary function. In the present paper we deal with the parametric family of boundaries $\{g_{a,b}(t)=c\sqrt{t+b}-a,b\ge 0,a>c\sqrt{b}\}$. 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 $n\to \infty$.

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