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

TL;DR

This paper derives bounds and asymptotic behaviors for the potential function of zero-mean, infinite variance random walks on integers, with applications to ladder height stability and escape probabilities.

Contribution

It provides new bounds and asymptotic formulas for the potential function of such random walks, extending understanding of their long-term behavior.

Findings

Bounds for the potential function $a(x)$ in terms of $x/m(x)$

Asymptotic relations for $a(x)$ as $x o o \infty$

Conditions for the convergence of exit probabilities and ladder height stability

Abstract

Let $S_n =X_1+\cdots +X_n$ be an irreducible random walk (r.w.) on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$. We obtain an upper and lower bounds of the potential function, $a(x)$, of $S_n$ in the form $a(x)\asymp x/m(x)$ under a reasonable condition on the distribution of $X_n$; we especially show that as $x\to\infty$ $$a(x) \asymp \frac{x}{m_-(x)} \quad\mbox{and}\quad \frac{a(-x)}{a(x)} \to 0 \quad\;\;\mbox{if}\quad \lim_{x\to +\infty} \frac{m_+(x)}{m_-(x)} =0,$$ where $m_\pm(x) = \int_0^xdy\int_y^\infty P[\pm X_1>u]du$ and $m=m_++m_-$. Under certain conditions on the tails of the distribution of $X$ we derive precise asymptotic forms of $a(x)$ as $x\to +\infty$ or/and $-\infty$. The results are applied to derive a sufficient condition for the relative stability of the ladder height and estimates of some escape probabilities from the origin; we show among others that under the above condition on $m_+/m-$, $P[S_n>0] \to 1/\alpha$ if and only if the probability of exiting a long interval $[-Q,R]$ through the upper boundary converges to $\lambda^{\alpha-1}$ as $Q/(Q+R) \to \lambda$ for any $0<\lambda<1$.