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

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 $0<\lambda<1$.