Geodesic
An asymptotic expansion for the distribution of the supremum of a random walk
Studia Mathematica, Tome 140 (2000) no. 1, pp. 41-55
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Let ${S_n}$ be a random walk drifting to -∞. We obtain an asymptotic expansion for the distribution of the supremum of ${S_n}$ which takes into account the influence of the roots of the equation $1-∫_ℝe^{sx}F(dx)=0,F$ being the underlying distribution. An estimate, of considerable generality, is given for the remainder term by means of submultiplicative weight functions. A similar problem for the stationary distribution of an oscillating random walk is also considered. The proofs rely on two general theorems for Laplace transforms.
DOI : 10.4064/sm-140-1-41-55
Keywords: random walk, supremum, submultiplicative function, characteristic equation, absolutely continuous component, oscillating random walk, stationary distribution, asymptotic expansions, Banach algebras, Laplace transform 
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M. S. Sgibnev. An asymptotic expansion for the distribution of the supremum of a random walk. Studia Mathematica, Tome 140 (2000) no. 1, pp. 41-55. doi: 10.4064/sm-140-1-41-55

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