Abstract
We investigate the maximum increment of a random walk with heavy-tailed jump size distribution. Here heavy-tailedness is understood as regular variation of the finite-dimensional distributions. The jump sizes constitute a strictly stationary sequence. Using a continuous mapping argument acting on the point processes of the normalized jump sizes, we prove that the maximum increment of the random walk converges in distribution to a Fréchet distributed random variable.
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We would like to thank the anonymous reviewers for useful comments that helped to improve the paper. Thomas Mikosch’s research is partly supported by the Danish Natural Science Research Council (FNU) Grant 09-072331, “Point process modelling and statistical inference” and 10-084172 “Heavy tail phenomena: Modeling and estimation”. Martin Moser would like to thank for the hospitality of the Department of Mathematics at the University of Copenhagen, where most parts of this work have been developed. Furthermore, he gratefully acknowledges the support of both, the Technische Universität München—Institute for Advanced Study, funded by the German Excellence Initiative, as well as the International Graduate School of Science and Engineering (IGSSE) at Technische Universität München, Germany.
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Mikosch, T., Moser, M. The limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes. Probab. Theory Relat. Fields 156, 249–272 (2013). https://doi.org/10.1007/s00440-012-0427-2
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DOI: https://doi.org/10.1007/s00440-012-0427-2
Keywords
- Maximum increment of a random walk
- Dependent jump sizes
- Moving average process
- GARCH process
- Stochastic volatility model
- Regular variation
- Extreme value distribution
Mathematics Subject Classification
- Primary 60G50
- 60G70
- Secondary 60F10