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Exact asymptotic behaviour of the distribution of the supremum

✍ Scribed by M.S. Sgibnev

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
139 KB
Volume
52
Category
Article
ISSN
0167-7152

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✦ Synopsis

Asymptotic expansions are obtained for the distribution of the supremum of a random walk with negative drift. The in uence of the roots of the characteristic equation is taken into account. The exact tail behaviour of the remainder terms is determined.

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We consider a random walk drifting to -∞ with distribution F of the steps. The paper considers the exact asymptotic behaviour of the distribution D of the supremum when there exists ¿ 0 such that R e x F(d x) = 1; R |x|e x F(d x) ‘ ∞ and R x 2 e x F(d x) = ∞, thus ÿlling the remaining gap in describ

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Let M~ be the supremum of a random walk drifting to -oo which is generated by the partial sums of a sequence of independent identically distributed random variables with a common distribution F. We prove that the moment generating function Eexp(sM~) is a rational function if and only if the function