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An asymptotic expansion for the distribution of the supremum of a random walk

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Sgibnev M. S.

Studia Mathematica

2000

tom 140

nr 1

41-55

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.

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1 Studia Mathematica

1 Sgibnev M. S.

1 2000

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An asymptotic expansion for the distribution of the supremum of a random walk