Overshoots over curved boundaries

Abstract

We consider the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n) :|x| ≤ rnb} as r → ∞. In order to be sure that this actually occurs, we treat only the case where the power b lies in the interval [0,½), and we further assume a condition that prevents the probability of exiting at either boundary tending to 0. Under these restrictions we establish necessary and sufficient conditions for the pth moment of the overshoot to be O(rq), and for the overshoot to be tight, as r → ∞.