## Tempered Stable Distributions: Stochastic Models for Multiscale ProcessesThis brief is concerned with tempered stable distributions and their associated Levy processes. It is a good text for researchers interested in learning about tempered stable distributions. A tempered stable distribution is one which takes a stable distribution and modifies its tails to make them lighter. The motivation for this class comes from the fact that infinite variance stable distributions appear to provide a good fit to data in a variety of situations, but the extremely heavy tails of these models are not realistic for most real world applications. The idea of using distributions that modify the tails of stable models to make them lighter seems to have originated in the influential paper of Mantegna and Stanley (1994). Since then, these distributions have been extended and generalized in a variety of ways. They have been applied to a wide variety of areas including mathematical finance, biostatistics,computer science, and physics. |

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### Contents

1 | |

5 | |

3 Tempered Stable Distributions | 15 |

4 Limit Theorems for Tempered Stable Distributions | 47 |

5 Multiscale Properties of Tempered Stable Lévy Processes | 67 |

6 Parametric Classes | 83 |

7 Applications | 97 |

8 Epilogue | 111 |

113 | |

117 | |

### Other editions - View all

Tempered Stable Distributions: Stochastic Models for Multiscale Processes Michael Grabchak No preview available - 2016 |

### Common terms and phrases

ˇ ˇ ˇ assume Borel function Borel set characteristic function cumulant define Definition denote density discuss distribution on Rd domain of attraction dominated convergence ETSp exists exponential extended Rosiński measure fact finite Borel measure finite measure Fix c e Gaussian given holds implies infinite variance stable infinitely divisible distributions integral jxj1 jxj2R.dx jxjÄ1 jZij Lemma Lévy flight Lévy measure lim sup limt!c Michael Grabchak 2016 models Monotone Convergence Theorem NRd0 open set p-tempered stable Lévy parameter polar decomposition probability measure process with XI proper p-tempered O-stable Proposition 2.6 R.dx R(dx Radon measures random variables regularly varying relatively compact risk-neutral risk-neutral measure Rosiniski satisfies sequence short time behavior ſº SpringerBriefs in Mathematics stable Lévy processes STLF tempered stable distributions vague convergence variance stable distribution weak convergence write