<p> We consider a one dimensional random walk in a random environment (RWRE) with a positive speed lim<i><sub>n</sub></i><sub>→∞</sub> (<i>X<sub>n</sub>/</i>) = υ<sub>α</sub> > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities <i>P</i><sub> ω</sub>(<i>X<sub>n</sub></i> < <i>xn</i>) with <i> x</i>∈ (0,υ<sub>α</sub>) decay approximately like exp{-<i> n</i><sup>1-1/</sup><i><sup>s</sup></i>} for a deterministic <i> s</i> > 1. More precisely, they showed that <i>n</i><sup> -γ</sup> log <i>P</i><sub>ω</sub>(<i>X<sub>n </sub></i> < <i>xn</i>) converges to 0 or -∞ depending on whether γ > 1 - 1/<i>s</i> or γ < 1 - 1/<i> s</i>. In this paper, we improve on this by showing that <i>n</i><sup> -1+1/</sup><i><sup>s</sup></i> log <i>P</i><sub> ω</sub>(X<sub>n</sub> < <i>xn</i>) oscillates between 0 and -∞ , almost surely.</p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:10170588 |
| Date | 28 October 2016 |
| Creators | Ahn, Sung Won |
| Publisher | Purdue University |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | Unknown |
| Type | thesis |
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