The Annals of Applied Probability

Complete corrected diffusion approximations for the maximum of a random walk

Jose Blanchet and Peter Glynn

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Consider a random walk (Sn:n≥0) with drift −μ and S0=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=maxn≥0Sn. 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, Eμexp(−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 EμSτ [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].

Article information

Ann. Appl. Probab. Volume 16, Number 2 (2006), 951-983.

First available in Project Euclid: 29 June 2006

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F05: Central limit and other weak theorems 62L10: Sequential analysis 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Corrected diffusion approximations random walks ladder heights sequential analysis single-server queue


Blanchet, Jose; Glynn, Peter. Complete corrected diffusion approximations for the maximum of a random walk. Ann. Appl. Probab. 16 (2006), no. 2, 951--983. doi:10.1214/105051606000000042.

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