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The article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster and then we apply the developed theory to determine the limiting…

Probability · Mathematics 2022-03-28 Bojan Basrak , Nikolina Milinčević , Petra Žugec

In many areas of interest, modern risk assessment requires estimation of the extremal behaviour of sums of random variables. We derive the first order upper-tail behaviour of the weighted sum of bivariate random variables under weak…

Statistics Theory · Mathematics 2022-08-17 Jordan Richards , Jonathan A. Tawn

In this paper we characterize the limiting behavior of sums of extreme values of long range dependent sequences defined as functionals of linear processes with finite variance. The extremal sums behave completely different by compared to…

Probability · Mathematics 2007-06-13 Rafal Kulik

In this paper we consider a heavy-tailed stochastic volatility model, $X_t=\sigma_tZ_t$, $t\in\mathbb{Z}$, where the volatility sequence $(\sigma_t)$ and the i.i.d. noise sequence $(Z_t)$ are assumed independent, $(\sigma_t)$ is regularly…

Statistics Theory · Mathematics 2013-12-11 Thomas Mikosch , Mohsen Rezapour

In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree,…

Probability · Mathematics 2022-04-07 Ayan Bhattacharya

To disentangle the complex non-stationary dependence structure of precipitation extremes over the entire contiguous U.S., we propose a flexible local approach based on factor copula models. Our sub-asymptotic spatial modeling framework…

Applications · Statistics 2019-03-26 Daniela Castro-Camilo , Raphaël Huser

The analysis of extremal dependence in high dimensions has recently attracted considerable interest. Existing methodology primarily focuses on modeling and estimation of extremal dependence structures, often supported by concentration…

Statistics Theory · Mathematics 2026-04-02 Axel Bücher , Yeonjoon Choi , Katharina Effertz , Stanislav Volgushev

In this paper we propose a framework that enables the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track not of the magnitude of the extreme…

Probability · Mathematics 2009-08-21 Henrik Hult , Gennady Samorodnitsky

We continue the study of the maximum of the scale-inhomogeneous discrete Gaussian free field in dimension two. In this paper, we consider the regime of weak correlations and prove the convergence in law of the centred maximum to a randomly…

Probability · Mathematics 2020-10-05 Maximilian Fels , Lisa Hartung

The impact of an extreme climate event depends strongly on its geographical scale. Max-stable processes can be used for the statistical investigation of climate extremes and their spatial dependencies on a continuous area. Most existing…

Methodology · Statistics 2023-06-14 Justus Contzen , Thorsten Dickhaus , Gerrit Lohmann

Identifying directions where extreme events occur is a major challenge in multivariate extreme value analysis. In this paper, we use the concept of sparse regular variation introduced by Meyer and Wintenberger (2021)} to infer the tail…

Statistics Theory · Mathematics 2023-01-09 Nicolas Meyer , Olivier Wintenberger

The extreme values theory presents specific tools for modeling and predicting extreme phenomena. In particular, risk assessment is often analyzed through measures for tail dependence and high values clustering. Despite technological…

Statistics Theory · Mathematics 2020-03-23 Helena Ferreira , Marta Ferreira

The aim of this paper is to provide models for spatial extremes in the case of stationarity. The spatial dependence at extreme levels of a stationary process is modeled using an extension of the theory of max-stable processes of de Haan and…

Statistics Theory · Mathematics 2007-06-13 Laurens de Haan , Teresa T. Pereira

In this note, we establish the convergence in distribution of the maxima of i.i.d. random variables to the Gumbel distribution with the associated normalizing sequences for several examples that are related to the normal distribution.…

Probability · Mathematics 2021-03-29 Markus Bibinger

A family of self-similar and translation-invariant random sup-measures with long-range dependence are investigated. They are shown to arise as the limit of the empirical random sup-measure of a stationary heavy-tailed process, inspired by…

Probability · Mathematics 2018-10-11 Olivier Durieu , Yizao Wang

The risk of catastrophes is related to the possibility of occurring extreme values. Several statistical methodologies have been developed in order to evaluate the propensity of a process for the occurrence of high values and the permanence…

Statistics Theory · Mathematics 2019-05-08 Helena Ferreira , Marta Ferreira

In this paper we consider the product of two positive independent risks $Y_1$ and $Y_2$. If $Y_1$ is bounded and $Y_2$ has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at…

Probability · Mathematics 2014-06-24 Krzysztof Dȩbicki , Julia Farkas , Enkelejd Hashorva

We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random.…

Probability · Mathematics 2026-03-10 Partha S. Dey , S. Rasoul Etesami , Aditya S. Gopalan

The generalized extreme value distribution and its particular case, the Gumbel extreme value distribution, are widely applied for extreme value analysis. The Gumbel distribution has certain drawbacks because it is a non-heavy-tailed…

Methodology · Statistics 2015-08-12 E. C. Pinheiro , S. L. P. Ferrari

We consider multivariate extreme value statistics for independent but nonidentically distributed random vectors. In particular, the data may have varying tail copulas and also heteroscedastic marginal distributions. Assuming smoothly…

Statistics Theory · Mathematics 2026-04-14 John H. J. Einmahl , Chen Zhou