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.
تحميل البحث