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Asymptotic expansions for first passage times

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  • Woodroofe, Michael

Abstract

Let F be a strongly non-lattice distribution function with a positive mean, a positive variance, and a finite third moment. Let X1, X2,... be i.i.d. with common distribution function F; and let Sn=X1+...+Xn, and ta = inf{n[greater-or-equal, slanted]1:Sn > a} for n [greater-or-equal, slanted] 1 and a >0. The main result reported here is a two term asymptotic expansion for Ha(n, z) = P{ta [infinity]. Assuming higher moments, a three term expansion for P{ta [less-than-or-equals, slant] n} and refined estimates for the probability of ruin in finite time are obtained as simple corollaries. A key tool is an asymptotic expansion in Stone's formulation of the local limit theorem.

Suggested Citation

  • Woodroofe, Michael, 1988. "Asymptotic expansions for first passage times," Stochastic Processes and their Applications, Elsevier, vol. 28(2), pages 301-315, June.
    • Handle: RePEc:eee:spapps:v:28:y:1988:i:2:p:301-315
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