Upper and lower space-time envelopes for oscillating random walks conditioned to stay positive

Abstract

We provide integral tests for functions to be upper and lower space time envelopes for random walks conditioned to stay positive. As a result we deduce a 'Hartman-Winter' Law of the Iterated Logarithm for random walks conditioned to stay positive under a third moment assumption. We also show that under a second moment assumption the conditioned random walk grows faster than n^½ (log n)^(-(1+e)) for any e > 0. The results are proved using three key facts about conditioned random walks. The first is the step distribution obtained in Bertoin and Doney (1994), the second is the pathwise construction in terms of excursions in Tanaka (1998) and the third is a new Skorohod type embedding of the conditioned process in a Bessel-3 process.