Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative

[摘要] We show that under a 3 + delta moment condition (where delta > 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 n(1/2) (log n)(-(1+epsilon)) for any epsilon > 0 and yet slower than n(1/2) (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. (C) 2003 Elsevier B.V. All rights reserved.

[发布日期] 2003-12-01

[关键词] random walk;conditioned random walk;Bessel process;Skorohod embedding;excursions;law of the iterated logarithm