We show that under a 3+δ moment condition (where δ>0) there exists a ‘Hartman–Winter’ Law of the iterated logarithm for random walks conditioned to stay non-negative. We also show that under a second moment assumption the conditioned random walk eventually grows faster than n1/2(log n)−(1+ε) for any ε>0 and yet slower than n1/2(log n)−1. The results are proved using three key facts about conditioned random walks. The first is the relation of its step distribution to that of the original random walk given by Bertoin and Doney (Ann. Probab. 22 (1994) 2152). The second is the pathwise construction in terms of excursions in Tanaka (Tokyo J. Math. 12 (1989) 159) and the third is a new Skorohod-type embedding of the conditioned process in a Bessel-3 process.
Hambly, B. M., Kersting, G., & Kyprianou, A. E. (2003). Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative. Stochastic Processes and their Applications, 108(2), 327-343. https://doi.org/10.1016/j.spa.2003.07.002