Complete Corrected Diffusion Approximations for the Maximum of Random Walk (with P. Glynn)

Summary:

This is the main chapter of my PhD dissertation. It revisits a paper of David Siegmund (Corrected Diffusion Approximations in Certain Random Walk Problems. Adv. Appl. Prob., 1979’). Siegmund provides a first order correction term (as the negative drift of the random walk tends to zero) to the Brownian approximation of the first passage time probability to a high positive level. A subsequent paper by Joseph Chang (Ann. App. Prob. 1996) adds another correction term and J. Chang and Y. Peres (Ann. Prob. 1997) find a full asymptotic expansion for Gaussian random walks. This paper obtains the full expansion for general light-tailed random walks (assuming strongly non-lattice distributions).

Bibtex:

@Article{BlaGlyCDA2006,

author = {J. Blanchet and P. Glynn.},

title = { Complete corrected diffusion for the maximum of the random walk.},

journal = {Annals of Applied Probability},

year = {2006},

volume = {16},

pages = {951-953}

}