Asymptotic expansions for first passage times

Michael Woodroofe

Stochastic Processes and their Applications, 1988, vol. 28, issue 2, 301-315

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.

Keywords: Edgeworth; expansions; local; limit; theorem; ruin; problems (search for similar items in EconPapers)
Date: 1988
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