## Abstract

There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions are shown to be equivalent. The introduction of a norm-like function, called modulus, yields a polar decomposition similar to the one in Euclidean spaces. The angular component of the time series, called angular or spectral tail process, captures all aspects of extremal dependence. The stationarity of the underlying series induces a transformation formula of the spectral tail process under time shifts.

Original language | English |
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Pages (from-to) | 539-566 |

Number of pages | 28 |

Journal | Extremes |

Volume | 20 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sept 2017 |

Externally published | Yes |

## Keywords

- Extremal index
- Extremogram
- Regular variation
- Spectral tail process
- Stationarity
- Tail dependence
- Time series