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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 J 1topology, 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 processes60G70: Extreme value theory; extremal processes

Information

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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