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Convergence to Stable Laws in the Space D

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

François Roueff  and
Philippe Soulier
François Roueff*
Affiliation:
Télécom ParisTech
Philippe Soulier*
Affiliation:
Laboratoire MODAL'X Université de Paris Ouest
*
∗ Postal address: Institut Mines-Télécom, Télécom ParisTech, CNRS LTCI-UMR5141, 46 rue Barrault, 75634 Paris Cedex 13, France.
∗∗ Postal address: Laboratoire MODAL'X Université de Paris Ouest, Nanterre, 92000, France. Email address: philippe.soulier@u-paris10.fr
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Abstract

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We study the convergence of centered and normalized sums of independent and identically distributed random elements of the space D of càdlàg functions endowed with Skorokhod's J1 topology, to stable distributions in D. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

Keywords

Regular variationstable processfunctional convergence

MSC classification

Primary: 60G52: Stable processes
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 1 - 17
Copyright
© Applied Probability Trust 

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