Notes on Random Walks in the Cauchy Domain of Attraction

Abstract : The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has been in many cases left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk-Nagaev inequality and a local version of it ; a large deviation theorem ; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems -- extending standard renewal theorems to the case of random walks. Most of our techniques carry through to the case of random walks in the domain of attraction of an $\alpha$-stable law with $\alpha \in(0,2)$, so we also present results in that case, since many of them seem to be missing in the literature.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01576409
Contributeur : Quentin Berger <>
Soumis le : mercredi 23 août 2017 - 10:01:02
Dernière modification le : vendredi 25 août 2017 - 01:10:34

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NotesCauchy_rev1.pdf
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  • HAL Id : hal-01576409, version 1
  • ARXIV : 1706.07924

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Quentin Berger. Notes on Random Walks in the Cauchy Domain of Attraction. 2017. <hal-01576409>

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