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

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 $a(x)\asymp x\mathrm{/}m(x)$ under a reasonable condition on the distribution of $X_n$; we especially show that as $x\to \mathrm{\infty }$ $$a(x)\asymp \frac{x}{m_{-}(x)}\text{quadand}\frac{a(-x)}{a(x)}\to 0if\underset{x\to +\mathrm{\infty }}{\lim }\frac{m_{+}(x)}{m_{-}(x)}=0,$$ where $m_{\pm }(x)={\int }_0^{x}dy{\int }_y^{\mathrm{\infty }}P[\pm X_1>u]du$ and $m=m_{+}+m_{-}$. Under certain conditions on the tails of the distribution of $X$ we derive precise asymptotic forms of $a(x)$ as $x\to +\mathrm{\infty }$ or/and $-\mathrm{\infty }$. 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 $m_{+}\mathrm{/}m-$, $P[S_n>0] \to 1/\alpha$ifandonlyiftheprobabilityofexitingalonginterval$[-Q,R]$throughtheupperboundaryconvergesto$\lambda^{\alpha-1}$as$Q/(Q+R) \to \lambda$forany$0<\lambda<1$.