Spatial extreme value theory and properties of max-stable processes Poitiers, November 8-10, 2012

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1 Spatial extreme value theory and properties of max-stable processes Poitiers, November 8-10, 2012 November 8, :00 Clement Dombry Habilitation thesis defense (in french) 17:00 Snack buet November 9, :00 Laurens de Haan Estimation of the marginal expected shortfall: the mean when a related variable is extreme 09:30 Gennady Samorodnitsky Functional Central Limit Theorem for Heavy Tailed Stationary Innitely Divisible Processes Generated by Conservative Flows 10:00 Philippe Soulier Weak convergence to stable processes in the space D 10:30 Coee break 11:00 Stilian Stoev Extreme value theory with operator norming 11:30 Ana Ferreira The generalized Pareto process (?); characterizations and properties 12:00 Lunch at Hotel Alteora 14:00 Philippe Naveau Analysis of heavy rainfall in high dimensions 14:30 Zakhar Kabluchko Zeros of the partition function in the Random Energy Model and Extreme Value Theory

2 15:00 Chen Zhou The integral of a stochastic process: tail probability and high quantile 15:30 Coee break 16:00 Mathieu Ribatet Conditional simulation of max-stable processes 16:30 Marco Oesting Representations of max-stable processes based on single extreme events November 10, :00 Possible free discussions in the seminar room of Hotel de l'europe.

3 Laurens DE HAAN, Erasmus University Rotterdam Estimation of the marginal expected shortfall: the mean when a related variable is extreme Joint work with J.J.Cai, J.H.J.Einmahl and C.Zhou. Denote the loss return on the equity of a nancial institution as X and that of the entire market as Y. For a given very small value of p > 0, the marginal expected shortfall (MES) is dened as E[X Y > Q Y (1 p)], where Q Y (1 p) is the (1-p)-th quantile of the distribution of Y. The MES is an important factor when measuring the systemic risk of nancial institutions. For a wide nonparametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p 0, as the sample size n. Since we are in particular interested in the case p = O(1/n), we use extreme value techniques for deriving the estimator and its asymptotic behaviour. The nite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large U.S. investment banks. Gennady SAMORODNITSKY, Cornell University Functional Central Limit Theorem for Heavy Tailed Stationary Innitely Divisible Processes Generated by Conservative Flows Joint work with Takashi Owada. We establish a functional central limit theorems for partial sum of certain symmetric stationary innitely divisible processes with regularly varying Lévy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments, that coincides, on a part of its parameter space, with a process previously described by Dombry and Guillotin-Plantard. The normalizing sequence and the limiting process are determined by the ergodic theoretical properties of the ow underlying the integral representation of the process, most importantly by its pointwise dual ergodicity. These properties can be interpreted as determining how long is the memory of the stationary innitely divisible process. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an innite measure. Philippe SOULIER, Université Paris Ouest Nanterre La Défense Weak convergence to stable processes in the space D We study the convergence of centered and normalized sums of i.i.d. random elements of the space D of càdlàg functions endowed with Skorohod's J 1 topology, to stable distributions in D. Our results are based on the concept of regular variation on metric spaces and on point process convergence. Stilian STOEV, University of Michigan Extreme value theory with operator norming Joint work with Mark M. Meerschaert and Hans-Peter Scheer. We propose a new approach to multivariate extremes using operator norming. We briey introduce some limit theorems for the angular extremes of multivariate heavy tailed data.

4 Operator norming allows us to handle in a unied way distributions with dierent tail exponents in dierent directions. We then present a method for simulating the limit process and a parametric bootstrap type procedure for testing for the need of operator norming. The statistical test is illustrated over simulated and real data sets. Ana FERREIRA, CEAUL, Lisbon University The generalized Pareto process (?); characterizations and properties We discuss a class of stochastic processes in the space of real continuous functions, verifying properties that generalize the ones known to characterize generalized Pareto random vectors (and variables). Philippe NAVEAU, CNRS, LSCE Saclay Analysis of heavy rainfall in high dimensions One of the main objectives of statistical climatology is to extract relevant information hidden in complex spatial-temporal climatological datasets. In impact studies, heavy rainfall are of primary importance for risk assessment linked to oods and other hydrological events. At an hourly time scale, precipitation distributions often strongly dier from Gaussianity. To identify spatial patterns, most well-known statistical techniques are based on the concept of intra and inter clusters variances (like the k-means algorithm or PCA's) and such approaches based on deviations from the mean may not be the most appropriate strategy in our context of studying rainfall extremes. One additional diculty resides in the dimension of climatological databases of hourly recordings that may gather measurements from hundreds or even thousands of weather stations during many decades. A possible avenue to ll up this methodological gap resides in taking advantage of multivariate extreme value theory, a well-developed research eld in probability, and to adapt it to the context of spatial clustering. In this talk, we propose and study two step algorithm based on this plan. Firstly, we adapt a Partitioning Around Medoids (PAM) clustering algorithm proposed by Kaufman to weekly maxima of hourly precipitation. This provides a set of homogenous spatial clusters of extremes of reasonable dimension. Secondly, we ne-tune our analysis by tting a Bayesian Dirichlet mixture model for multivariate extremes within each cluster. We compare and discuss our approach throughout the analysis of hourly precipitation recorded in France (Fall season, 92 stations, ). Zakhar KABLUCHKO, Ulm University Zeros of the partition function in the Random Energy Model and Extreme Value Theory The partition function of the Random Energy Model at inverse temperature β is given by Z N (β) = 2 n k=1 e β nx k, where X 1, X 2,... are i.i.d. standard normal random variables. We describe the structure of the complex zeros of Z N (β) as N thereby proving rigorously some predictions

5 of Derrida. Important role in this description is played by the extreme values in the sequence X 1, X 2,.... Chen ZHOU, Economics and research division, De Nederlandsche Bank The integral of a stochastic process: tail probability and high quantile Let X = X(s) s S be an almost sure continuous stochastic process (S compact subset of R d ) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of X(s)ds. Our methods are applied to the S total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall. We answer two questions based on daily rainfall observations over 30 years: rstly, what is the level of an once per 100-year total rainfall? Secondly, what is the probability that the total rainfall exceeds a high threshold? The rst question is investigated under a parametric model. The estimated level is then used as the threshold in the second question without assuming parametric models. The answer to the second question thus justies the parametric choice in the rst question. Mathieu RIBATET, Université de Montpellier Conditional simulation of max-stable processes Joint work with C.Dombry and F.Eyi-Minko. Since many environmental processes are spatial in extent, a single extreme event may aect several locations, and their spatial dependence has to be appropriately taken into account. This paper proposes a framework for conditional simulation of max-stable processes and gives closed forms for the regular conditional distributions of Brown Resnick and Schlather processes. We test the method on simulated data and give an application to extreme rainfall around Zurich and extreme temperatures. The proposed framework provides accurate conditional simulations and can handle real-sized problems. Marco OESTING, Mannheim University Representations of max-stable processes based on single extreme events Joint work with S. Engelke, A. Malinowski (Göttingen University) and M. Schlather (Mannheim University) This talk provides the basis for new methods of inference for max-stable processes ξ that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and, hence, will rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are presented. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identied.