Statistica Sinica 23 (2013), 1697-1715

BOUNDARY CROSSING DISTRIBUTIONS OF RANDOM

WALKS RELATED TO THE LAW OF THE ITERATED

LOGARITHM

Ulrich Kerkhoff and Hans Rudolf Lerche

University of Freiburg

To David Siegmund on his 70th birthday.

Abstract: A result for the first passage densities of Brownian motion as $t\to\infty$ was given in [#!16!#] for boundaries that grow faster than $\sqrt{t}$ as $t\to\infty$. 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.