Complete corrected diffusion approximations for the maximum of a random walk

Abstract

Consider a random walk (S n : n ≥ U) with dritt -μ and S 0 = 0. Assuming that the increments have exponential moments, negative mean, and are strongly nonlattice, we provide a complete asymptotic expansion (in powers of μ ≥ 0) that corrects the diffusion approximation of the all time maximum M = max n≥0 S n. Our results extend both the first-order correction of Siegmund [Adv. in Appl. Probab. 11 (1979) 701-719] and the full asymptotic expansion provided in the Gaussian case by Chang and Peres [Ann. Probab. 25 (1997) 787-802]. We also show that the Cramér-Lundberg constant (as a function of μ) admits an analytic extension throughout a neighborhood of the origin in the complex plane C. Finally, when the increments of the random walk have nonnegative mean μ, we show that the Laplace transform, E μ exp(-bR(∞)), of the limiting overshoot, R(∞), can be analytically extended throughout a disc centered at the origin in ℂ × ℂ (jointly for both b and μ). In addition, when the distribution of the increments is continuous and appropriately symmetric, we show that E μS τ [where τ is the first (strict) ascending ladder epoch] can be analytically extended to a disc centered at the origin in ℂ, generalizing the main result in [Ann. Probab. 25 (1997) 787-802] and extending a related result of Chang [Ann. Appl. Probab. 2 (1992) 714-738]. © Institute of Mathematical Statistics, 2006.