Asymptotic Expansions for the Distributions of Stopped Random Walks and First Passage Times

Let S n = X 1 +...X n , n ≥1, be a d-dimensional random walk and let T a = inf{n ≥n a : ng(S n /n) ≥ a), where n a = o(a). Let θ=g(Ex 1 ), θ n = g(S n /n) and Δ a =T a θ Ta -a. Edgeworth-type expansions are developed for P{Ta = n, y 1 ≤ Δ a ≤ y 2 } and for the distribution functions of T a and of √Ta(h(θ Ta )-h(θ)), where h is a real-valued function such that h'(θ) ? 0.