Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk

In other words, X does not have exponential moments. We say that X has light tails, if E exp (θ |X|) <∞ for some θ > 0. The distribution of Mδ is of importance in a number of different disciplines. For x > 0, {Mδ > x} = {τ δ (x) < ∞}, where τ δ (x) = inf{n ≥ 1 : S n > x}, so that computing the tail of Mδ is equivalent to computing a level crossing probability for the random walk S. Because of this level crossing interpretation, the tail of Mδ is of great interest to both the sequential analysis and risk theory communities. In particular, in the setting of insurance risk, P (τ δ (x) <∞) is the probability that an insurer will face ruin in finite time (when the insurer starts with initial reserve x and is subjected to iid claims over time); see, for example, Asmussen (2001). The distribution of Mδ also arises in the analysis of the single most important model in queueing theory, namely the single-server queue. If the inter-arrival and service times for successive customers are iid with a mean arrival rate less than the mean service rate, then W = (Wn : n ≥ 0) is a positive recurrent Markov chain