The potential function and ladder heights of a recurrent random walk on $\mathbb {Z}$ with infinite variance

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We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a(x)$ to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin.

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