Inequalities for first-exit probabilities and expected first-exit times of a random walk

For the partial sum process of independent identically distributed random variables we derive sequences of new upper and lower bounds for the probability that it will exit from a given compact interval to the left and for the expected first-exit time. The sequences are generated by the same iterative procedure starting from different initial values. They converge monotonically to the desired first-exit quantities at an exponential rate.