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 $m_{\pm }(x)={\int }_0^{x}dy{\int }_y^{\infty }P[\pm X_1>u]du$ 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 $0<\lambda <1$.

60 pages, A new theorem (Theorem 7 in the revised version) is added

References in corpus (1)

• Asymptotic behaviour of a random walk killed on a finite set

Cited by in corpus (4)

• The two-sided exit problem for a random walk on $\mathbb{Z}$ and having infinite variance II
• The two-sided exit problem for a random walk on $\mathbb{Z}$ with infinite variance I
• Recurrent random walks on $\mathbb{Z}$ with infinite variance: transition probabilities of them killed on a finite set
• The potential function and ladder variables of a recurrent random walk on $\mathbb{Z}$ with infinite variance