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

Kohei Uchiyama

Published 2018-02-27Version 1

We consider an irreducible random walk on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$ and obtain certain asymptotic properties of the potential function, $a(x)$, of the walk; in particular we 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 $m_\pm(x) = \int_0^xdy\int_y^\infty P[\pm X_1>u]du$. The results are applied in order to obtain a sufficient condition for the relative stability of the ladder height and estimates of some escape probabilities from the origin.