Abstract
We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64, 193–208, 1969) and Nagaev (Ann Probab 7:745–789, 1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (Stoch Proc Appl 44:291–327, 1993; 68:1–20, 1997) in the context of central limit theorems with infinite variance stable limits. We illustrate the principle for stochastic volatility models, real valued functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.
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T. Mikosch’s research is partly supported by the Danish Research Council (FNU) Grants 272-06-0442 and 09-072331. The research of T. Mikosch and O. Wintenberger is partly supported by a Danish-French Scientific Collaboration Grant of the French Embassy in Denmark. Both authors would like to thank their home institutions for hospitality when visiting each other.
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Mikosch, T., Wintenberger, O. Precise large deviations for dependent regularly varying sequences. Probab. Theory Relat. Fields 156, 851–887 (2013). https://doi.org/10.1007/s00440-012-0445-0
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DOI: https://doi.org/10.1007/s00440-012-0445-0
Keywords
- Stationary sequence
- Large deviation principle
- Regular variation
- Markov processes
- Stochastic volatility model
- GARCH