### Stochastic Models with Heavy Tails

**Thomas Mikosch (Copenhagen University)**

We interpret ``heavy tails'' in the sense of regular variation of the univariate and multivariate distributions in a sequence of variables or vectors, typically assumed to be strictly stationary. Regularly varying distributions are suitable for large claims data in (re)insurance, log-return data in finance, various quantities in telecommunications such as file sizes and transmission durations of files. This type of distributions is also popular for extreme weather modeling. Distributions with regularly varying tails characterize domains of attraction for sums and maxima of iid and weakly dependent sequences with infinite variance and Fréchet limits, respectively. Beyond this characterization, they are flexible models in various applied contexts where the domain of attraction aspect is less relevant. For example, the Kesten-Goldie theory for stochastic recurrence equations yields univariate and multivariate power-law tails for the stationary solution of such equations. We will give some reasons for the origins of this tail behavior and some applications to financial return modeling (GARCH processes).

The Kesten-Goldie class of stationary processes with power-law tails is just one element in the class of regularly varying stationary sequences. Historically, linear processes with regularly varying noise were among the first time series models with heavy tails. By now, many more time series models with this property are in our toolbox such as regularly varying stochastic volatility and GARCH processes as well as models arising from non-affine stochastic recurrence equations. Under mild dependence conditions, the class of regularly varying processes shares various asymptotic properties with a regularly varying iid sequence: maxima converge to Fréchet distributed limits, the processes of the scaled points of the time series converge weakly to a (compound) Poisson process, Nagaev-type large deviations hold and infinite variance stable limit theory applies.

Nagaev-type large deviations lie at the heart of many heavy-tail limit results when partial sums are involved, for example in ruin theory for a random walk with negative drift. A more recent application is in the theory of random matrices, in particular when considering sample (auto-)covariance matrices for regularly time series whose dimension increases with the sample size. We will consider such models and explain how heavy tails manifest in the spectral analysis of the aforementioned matrices.

The four lecture will deal with the following topics (tentatively):

- Notions of heavy-tailedness. Univariate and multivariate regular variation for random variables and vectors. Regularly varying sequences.Regularly varying structures under mappings.

- Examples of regularly varying sequences. Origins of regular variation in a time series. An excursion to the Goldie-Kesten theory. Asymptotic results for regularly varying sequences.

- Nagaev-type large deviations for regularly varying sequences. Applications in asymptotic and ruin theory.

- Regularly varying time series with increasing dimension. Eigenvalues and eigenvectors of their sample (auto-)covariance matrices.

Date | Course | Place |
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01 Jun 2016 10:00 | Stochastic Models with Heavy Tails | Université Pierre et Marie Curie, Room 16-26-113 |

02 Jun 2016 10:00 | Stochastic Models with Heavy Tails | Université Pierre et Marie Curie, Room 16-26-113 |

08 Jun 2016 10:00 | Stochastic Models with Heavy Tails | Université Pierre et Marie Curie, Room 16-26-113 |

09 Jun 2016 10:00 | Stochastic Models with Heavy Tails | Université Pierre et Marie Curie, Room 16-26-113 |