Abstract: A result for the first passage densities of Brownian motion aswas given in [#!16!#] for boundaries that grow faster than
as
. From this result the Kolmogorov-Petrovski-Erdos test near infinity has been derived. Here we extend these results to first passage probabilities of random walks. The asymptotic formulas are the same as for Brownian motion and, especially, no overshoot term shows up.
Key words and phrases: Curved boundary crossing of random walks, overshoot calculations, stopping times and the law of the iterated logaritm.