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