Complete corrected diffusion approximations for the maximum of a random walk

Consider a random walk $(S_n:n\geq0)$ with drift $-\mu$ 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 $\mu>0$) that corrects the diffusion approximation of the all time maximum $M=\max_{n\geq0}S_n$. 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 Cram\'{e}r--Lundberg constant (as a function of $\mu$) admits an analytic extension throughout a neighborhood of the origin in the complex plane $\mathbb{C}$. Finally, when the increments of the random walk have nonnegative mean $\mu$, we show that the Laplace transform, $E_{\mu}\exp(-bR(\infty))$, of the limiting overshoot, $R(\infty)$, can be analytically extended throughout a disc centered at the origin in $\mathbb{C\times C}$ (jointly for both $b$ and $\mu$). In addition, when the distribution of the increments is continuous and appropriately symmetric, we show that $E_{\mu}S_{\tau}$ [where $\tau$ is the first (strict) ascending ladder epoch] can be analytically extended to a disc centered at the origin in $\mathbb{C}$, 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].