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

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.

17 pages, 1 figure

References in corpus (1)

• One idea and two proofs of the KMT theorems

Cited by in corpus (1)

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