Geodesic
An asymptotic expansion for the distribution of the supremum of a random walk
Studia Mathematica, Tome 140 (2000) no. 1, pp. 41-55

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

DOI

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 
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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