An asymptotic expansion for the distribution of the supremum of a random walk
Volume 140 / 2000
Studia Mathematica 140 (2000), 41-55
DOI: 10.4064/sm-140-1-41-55
Abstract
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.
