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Stable limits for sums of dependent infinite variance random variables
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  • Published: 11 March 2010

Stable limits for sums of dependent infinite variance random variables

  • Katarzyna Bartkiewicz1,
  • Adam Jakubowski1,
  • Thomas Mikosch2 &
  • …
  • Olivier Wintenberger3 

Probability Theory and Related Fields volume 150, pages 337–372 (2011)Cite this article

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  • 38 Citations

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Abstract

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland

    Katarzyna Bartkiewicz & Adam Jakubowski

  2. Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Thomas Mikosch

  3. Centre De Recherche en Mathématiques de la Décision UMR CNRS 7534, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    Olivier Wintenberger

Authors
  1. Katarzyna Bartkiewicz
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  2. Adam Jakubowski
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  3. Thomas Mikosch
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  4. Olivier Wintenberger
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Corresponding author

Correspondence to Adam Jakubowski.

Additional information

Thomas Mikosch’s research is partly supported by the Danish Research Council (FNU) Grants 272-06-0442 and 09-072331. The research of Thomas Mikosch and Olivier Wintenberger is partly supported by a Scientific Collaboration Grant of the French Embassy in Denmark.

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Bartkiewicz, K., Jakubowski, A., Mikosch, T. et al. Stable limits for sums of dependent infinite variance random variables. Probab. Theory Relat. Fields 150, 337–372 (2011). https://doi.org/10.1007/s00440-010-0276-9

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  • Received: 27 August 2009

  • Revised: 06 February 2010

  • Published: 11 March 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0276-9

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Keywords

  • Stationary sequence
  • Stable limit distribution
  • Weak convergence
  • Mixing
  • Weak dependence
  • Characteristic function
  • Regular variation
  • GARCH
  • Stochastic volatility model
  • ARMA process

Mathematics Subject Classification (2000)

  • 60F05
  • 60G52
  • 60G70
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