Stable limits for sums of dependent infinite variance random variables

Katarzyna Bartkiewicz; Adam Jakubowski; Thomas Mikosch; Olivier Wintenberger

hal-00395525, version 4

Stable limits for sums of dependent infinite variance random variables

Katarzyna Bartkiewicz ¹, Adam Jakubowski ¹, Thomas Mikosch ², Olivier Wintenberger () ³

Probability Theory and Related Fields 150, 3 (2011) 337-372

Résumé : 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 ex- pressed 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) pro- cess and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.

1 : Faculty of Mathematics and Computer Science
Nicolaus Copernicus University, Torun
2 : Laboratory of Actuarial Mathematics
University of Copenhagen
3 : CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
CNRS : UMR7534 – Université Paris IX - Paris Dauphine

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Contributeur : Olivier Wintenberger <>
Soumis le : Vendredi 26 Mars 2010, 19:12:34
Dernière modification le : Mercredi 19 Octobre 2011, 09:57:11

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arXiv : 0906.2717

DOI : 10.1007/s00440- 010-0276-9

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%% hal-00395525, version 4
%% http://hal.archives-ouvertes.fr/hal-00395525
@article{bartkiewicz:hal-00395525,
    hal_id = {hal-00395525},
    url = {http://hal.archives-ouvertes.fr/hal-00395525},
    title = {{Stable limits for sums of dependent infinite variance random variables}},
    author = {Bartkiewicz, Katarzyna and Jakubowski, Adam and Mikosch, Thomas and Wintenberger, Olivier},
    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 ex- pressed 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) pro- cess and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.}},
    keywords = {stationary sequence, stable limit distribution, weak conver- gence, mixing, weak dependence, characteristic function, regular variation, GARCH, stochastic volatility model, ARMA process.},
    language = {Anglais},
    affiliation = {Faculty of Mathematics and Computer Science , Laboratory of Actuarial Mathematics , CEntre de REcherches en MAth{\'e}matiques de la DEcision - CEREMADE},
    pages = {337-372},
    journal = {Probability Theory and Related Fields},
    volume = {150},
    number = {3 },
    note = {35 pages Bourse de collaboration pour la recherche universitaire de l'ambassade de France au Danemark },
    audience = {internationale },
    doi = {10.1007/s00440- 010-0276-9 },
    year = {2011},
    month = Aug,
    pdf = {http://hal.archives-ouvertes.fr/hal-00395525/PDF/ArxivVersion.pdf},
}

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