Complete corrected diffusion approximations for the maximum of a random walk

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dc.creatorBlanchet, Jose
dc.creatorGlynn, Peter
dc.date2006-07-05
dc.date.accessioned2026-07-07T07:18:03Z
dc.date.available2026-07-07T07:18:03Z
dc.descriptionConsider a random walk $(S_n:n\geq0)$ with drift $-μ$ 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 $M=\max_{n\geq0}S_n$. 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 Cramér--Lundberg constant (as a function of $μ$) 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 $μ$, we show that the Laplace transform, $E_μ\exp(-bR(\infty))$, 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 $μ$). 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 $\mathbb{C}$, 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].
dc.descriptionPublished at http://dx.doi.org/10.1214/105051606000000042 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0607121
dc.identifierhttp://arxiv.org/abs/math/0607121
dc.identifierAnnals of Applied Probability 2006, Vol. 16, No. 2, 951-983
dc.identifierdoi:10.1214/105051606000000042
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/114162
dc.subjectProbability
dc.subject60G50 (Primary) 60F05, 62L10, 68M20 (Secondary)
dc.titleComplete corrected diffusion approximations for the maximum of a random walk
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