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

B M Hambly, G Kersting, A E Kyprianou

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)327-343
Number of pages17
JournalStochastic Processes and their Applications
Volume108
Issue number2
DOIs
Publication statusPublished - Dec 2003

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