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arXiv · 2010.08767

Bounds on the running maximum of a random walk with small drift

Ofer Busani·Timo Seppäläinen
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Abstract

We derive a lower bound for the probability that a random walk with i.i.d.\ increments and small negative drift $\mu$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|\mu| \log N)$. The approach is elementary and does not use strong approximation theorems.

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Ofer Busani, Timo Seppäläinen. 2020-10-17. Bounds on the running maximum of a random walk with small drift. https://arxiv.org/abs/2010.08767

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