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A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

Published online by Cambridge University Press:  09 April 2009

Paul Embrechts  and
John Hawkes
Paul Embrechts
Affiliation:
Departement Wiskunde KUL, Celestijnenlaan 200-B, B-3030 Heverlee, Belgium
John Hawkes
Affiliation:
University College of Swansea, Department of Statistics, Singleton Park, Swansea SA2 8PP, United Kingdom
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Abstract

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Suppose that (pn) is an infinitely divisible distribution on the non-negative integers having Lévy measure (vn). In this paper we derive a necessary and sufficient condition for the existence of the limit limn→∞ pn/vn. We also derive some other results on the asymptotic behaviour of the sequence (Pn) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index α, α ∈ (1,2).

Keywords

infinitely divisible sequencesLévy measuredomain of attractionfluctuation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 32 , Issue 3 , June 1982 , pp. 412 - 422
Copyright
Copyright © Australian Mathematical Society 1982

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