UCP, site St Martin, 2 av. Adolphe Chauvin, 95300 Pontoise ; Bât E 5^{ème} étage, salle de séminaires de mathématiques (5.54).

**ENTREE LIBRE, PAS DE FRAIS D’INSCRIPTION**

Durée indicative |
5 sept |
6 sept |
10 sept |
11 sept |

10:00-12:00 |
O. Wintenberger |
A. Jakubowski |
J. Segers |
J. Segers |

14:00-17:30 |
O. Wintenberger |
A. Jakubowski |
A. Janssen / H. Drees |
H. Drees / A. Janssen |

**Olivier Wintenberger (Paris IX Dauphine) : Limit theorems for dependent regularly varying partial sums**

In this short course, we develop new tools as the cluster index to characterize the limiting behavior of dependent regularly varying partial sums. Considering functions of regenerative Markov chains, the classical Nummelin's scheme gives direct interpretations in terms of independent sums of blocks of random length. As in the classical Central Limit Theorem, the geometric ergodicity is not sufficient. We use new drift conditions under which the \alpha-stable limits and the large deviations are characterized. Several examples are treated in details such as random affine equations solutions, GARCH(1,1) models...

**Adam Jakubiwski (Torun): Single sequence methods in limit theory for extremes**

**Johan Segers (Louvain la Neuve) : **
**Extremes of weakly dependent stationary sequences: clusters, tail processes, and point process convergence**

Extreme-value asymptotics of a stationary sequence of random variables are governed by both the global and the local dependence structure of the sequence. In case of weak dependence, the sequence may be divided into approximately independent and identically distributed blocks. As in the case of independent random variables, this division into blocks induces a Poisson character for the occurrence of extremes. However, if the dependence within blocks is left unrestricted, extremes may arrive in batches or clusters, yielding compound Poisson limits.

The distinctive feature of extremes of weakly dependent stationary sequences is therefore the appearance of clusters of extremes. The distribution of such clusters can be described via cluster functionals or cluster processes.

Stationarity of the original sequence implies that the asymptotic distribution of such clusters is fully determined by the conditional distribution of the process given that at a specific time point, an extreme value is produced. The limiting conditional distribution of the original sequence given such an event is called the tail process. For univariate Markov chains, the tail process takes the form of a random walk.

**Holger Drees (Hambourg) holger.drees@uni-hamburg.de :** Extremal serial dependence of time series

Modeling the dependence between consecutive observations in a time series plays a crucial role in risk management. For example, the risk of large losses from a financial investment is increased if extreme negative returns tend to occur in clusters, and heavy rainfall on several consecutive days could trigger a catastrophic flooding.

We will recall the so-called coeffcient of tail dependence (introduced by Ledford and Tawn, 1996) as an important measure of the strength of serial dependence between extremes. A general class of empirical processes introduced by Drees and Rootzén (2010) enables us to analyze the asymptotic behavior of estimators of the coefficient of tail dependence in a unified framework. Bootstrap versions of these empirical processes yield asymptotic confidence intervals.

In an application it is shown how to use these results to discriminate between time series of GARCH-type and time series with light-tailed stochastic volatilities and heavy-tailed innovations. An analysis of a time series of returns of the German blue stocks index however reveals that probably none of these time series models describe the extremal dependence structure accurately.

We will then introduce a new type of stochastic volatility model which allows for a more flexible dependence structure and thus may fit the data better in this respect.

(The last part of the talk is based on a joint project with Anja Janssen,University of Hamburg.)

References:

- Ledford, A.W., and Tawn, J.A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169-187.

- Drees, H., and Rootzén, H. (2010). Limit Theorems for Empirical Processes of Cluster Functionals. Ann. Statist. 38, 2145-2186.

**Anja Janßen (Hambourg) :** main topics: **hidden regular variation, index of regular variation of random difference equations**