Diffusion Approximations for the Maximum of a Per- Turbed Random Walk

Consider a random walk S = (Sn : n ≥ 0) that is “perturbed” by a stationary sequence (ξn : n ≥ 0) to produce the process S = (Sn + ξn : n ≥ 0). This paper is concerned with developing limit theorems and approximations for the distribution of Mn = max{Sk + ξk : 0 ≤ k ≤ n} when the random walk has a drift close to zero. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walk and extreme value limit theory.