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Main Course:
Main Course:
Prof. Thomas Mikosch (University of Copenhagen).
Abstract: In this course we give an introduction to the extreme value theory for time series models, i.e., models with serial dependence. Emphasis is on time series models with heavy tails. One finds examples of real-life time series of this kind in insurance, finance, seismology and telecommunica- tions. They are characterized by clusters of oberservations whose values exceed the normal range by far, and they can be very costly. Then classical tools like the autocorrelation function and spectral analysismare not suitable for describing the dependence structure in such time series.
We focus on regularly varying time series. This means that the tails of their marginal and finite-dimensional distributions are of power- law type. Early on, in the 1980s, linear models (ARMA) with regularly varying innovations were studied extensively by Holger Rootzén, Richard Davis and Sid Resnick. Although linear models are the backbone of classical time series analysis their extremes do not describe extremal clusters as oberved in financial return series. For these phenomena the class of stochastic volatility models is more adequate. Time series models relevant for finance were developed by Robert Engle (ARCH), Tim Bollerslev (GARCH) and others, starting in the 1980s. The extremal dynamics of these time series is totally different from heavy-tailed linear models. The latter ones are driven by heavy-tailed innovations, while the marginal distribution of ARCH-GARCH-type models have power-law tails generated by random coefficient difference equations whose innovations may be light-tailed. The power-law tails of these models were described in a seminal paper by Harry Kesten in 1973.
In the course, we will give an introduction to the notion of regular variation, starting from power-law type behavior of the tails of random variables. In the next step we extend regular variation to random vectors. In turn, we can define regular variation of real- or vector-valued time series by assuming that their finite-dimensional distributions are regularly varying. In 1995, Richard Davis and Tailen Hsing introduced the notion of a regularly varying time series. Several years later, in 2009, Bojan Basrak and Johan Segers characterized the extremal behavior of such time series by introducing the spectral tail process. It describes the propogation of extreme values through a serially dependent sequence into the past and future. Moreover, the spectral tail process allows one to describe the extremal cluster behavior of such time series in suitable ways, e.g. by considering a measure of the strength of extremal clustering (the extremal index) or an analog of the autocorrelation function for extremes (the extremogram).
We will give a short introduction to the extreme value theory of time series, showing the parallels and differences between sequences of independent and serially dependent variables. We illustrate the theory by considering a variety of time series models, including the benchmark of a linear process, solutions to stochastic recurrence equation (closely related to the ARCH-GARCH model), stochastic volatility models, max-stable processes. We touch on the relations between extreme value theory and point process theory. The latter one is a practical tool for describing extreme values in time and space. In particular, point processes are well suited for describing extremal clustering behavior. If time allows, we will also consider the influence of extreme values on the partial sums of time series. This topic is of particular interest when second moments are infinite. Then the extreme terms in a sample have strong influence on the behavior of the partial sums. Another phenomenon where the extremes have influence on the sums is large deviation probabilities: then the tails of sums at high thesholds are shaped by the structure of the spectral tail process.
This course is based on the monograph:
T. Mikosch and O. Wintenberger (2024) Extreme Value Theory for Time Series. Models with Power-Law Tails. Springer, New York.
Course material:
Accompanying Lectures
Accompanying Lectures
Geometric extremal graphical models - Prof. Jennifer Wadsworth (Lancaster University).
Abstract: A geometric representation for multivariate extremes, based on the shapes of scaled sample clouds in light-tailed margins and their so-called limit sets, has recently been shown to connect several existing extremal dependence concepts. Furthermore, this geometric representation provides a natural way to describe complex extremal dependence structures, which more established approaches to multivariate extremes do not represent well. These attractive properties have led to recent work that exploits the geometric approach as a foundation for statistical modelling, which has been demonstrated in relatively low dimensions thus far. For higher dimensional modelling, we require principled simplifications of the model structure. We will introduce the concept of geometric extremal graphical models, and outline some theoretical results based on block graphs. On the practical side, we will demonstrate some initial results employing these ideas to model joint river flows in the northwest of England. Based on joint works with Ioannis Papastathopoulos, and Kristina Grolmusova and Thordis Thorarinsdottir.
Domain-scaled regular variation - Mathematical foundations for a new tail approximation - Dr. Kirstin Strokorb (University of Bath).
Abstract: The concept of regular variation has a long history in probability and statistics. In the context of regularly varying spatial stochastic processes part of the mathematical appeal stems from the equivalence of multiple characterizations of regular variation, which in turn can be exploited for statistical inference and tail extrapolation. On the other hand, the asymptotic stability properties of a regularly varying process are rather rigid and are appropriate only for the modelling of the tail behaviour of so-called asymptotically dependent stochastic processes. Instead, empirical evidence suggests that spatial extremes often exhibit weakening of dependence that is not compatible with pure asymptotic dependence, that is, they become more localized as the conditioning threshold increases. In this talk I will introduce a refined notion of regular variation that overcomes this limitation via domain-scaling. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown–Resnick process.
We shall see that this modification via domain-scaling can conveniently be incorporated into the theory of regular variation for spatial stochastic processes to make it more flexible and better reflect empirical evidence. We study key properties of the resulting tail approximating process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness. This talk is based on joint work with Marco Oesting and Raphaël de Fondeville (preprint https://doi.org/10.14760/OWP-2025-02).
Self-normalization of sums of dependent random variables - Prof. Olivier Wintenberger (LPSM, Sorbonne University).
Abstract: In this talk, we analyze the extremal dependence for regularly varying stationary time series. After introducing the extremogram as a specific tool to visualize the length of memory for extreme values, we study the joint limit behavior of sums, maxima, and standard deviations of samples. As a consequence, we can determine the distributional limits for ratios of sums and maxima, as well as studentized sums, even when the variance is infinite. We calculate moments of the limits and point out the differences between the iid (independent and identically distributed) and some other weakly dependent time series. We highlight some unknown side effects of the clustering of extremes in common ratio statistics.
Based on a collaboration with M. Matsui (Osaka) and T. Mikosch (Copenhagen)
Contributed Talks
Contributed Talks
Causal tail coefficient for compound extremes in multivariate time series - Cathy Yin (Imperial College London)
Abstract: Extreme events are often multivariate in nature. A compound extreme occurs when a combination of variables jointly produces a significant impact, even if individual components are not necessarily marginally extreme. Compound extremes have been observed across a wide range of domains, including space weather, climate, and environmental science. For example, heavy rain-fall sustained over consecutive days can impose cumulative stress on urban drainage systems, potentially resulting in flooding. However, most existing methods for detecting extremal causality focus primarily on individual extreme values and lack the flexibility to capture causal relationships between compound extremes. This work introduces a novel framework for detecting causal dependencies between extreme events, including compound extremes. We define the compound causal tail coefficient that captures the extremal dependence of compound events between pairs of stationary time series. Based on a consistent estimator of this coefficient, we develop a bootstrap hypothesis test to evaluate the presence and direction of causal relationships. Our method can accommodate nonlinearity and latent confounding variables. We demonstrate the effectiveness of our method through simulations and an application to space-weather data.
The Tail That Wasn’t: Rethinking Extreme Value Theory for Real Data - Marwan Wehaiba El Khazen (Inria Paris)
Abstract: Extreme Value Theory (EVT) is one of the main mathematical tools for estimating rare-event probabilities, yet its practical use often depends on assumptions that real data do not visibly justify. This work argues that the central difficulty is not merely statistical scarcity, but tail representativity: the largest observed values may not belong to the mechanism that ultimately governs the true extreme region. In such cases, EVT fitted to the apparent tail estimates the wrong extremes. Moreover, real data are rarely independent or even stationary. Ruptures, outliers, regime changes, measurement artifacts, and genuine extremes can look statistically similar, while EVT is highly sensitive to their classification.
We propose a more conservative and mechanism-aware framework for EVT. First, EVT should be preceded by diagnostics of domain-of-attraction plausibility, tail convergence, dependence, and stationarity. Second, observed data should be segmented into approximately stationary regimes, with explicit algorithms distinguishing outliers, ruptures, and genuine extremes. Local extreme-value models may then be fitted within these regimes and merged into a global conservative extreme distribution, rather than forcing a single model onto a heterogeneous sample. We show that this framework not only produces estimations, but is able to discuss their validity and rethink future measurement itself.
We use this framework on simulated data that presents the aforementioned difficulties, and then on real data, in the context of measurement-based probabilistic timing analysis and probabilistic worst-case execution time estimation. We discuss the results and argue for a measurement-oriented coding hygiene: decomposing programs into meaningful subprograms, measuring cleaner components, deriving conservative bounds between decomposed and undecomposed executions using static analysis, and applying EVT only where its assumptions are defensible. The resulting hybrid framework can be validated on programs with known worst-case execution times.
Block maxima method by the tail: a hybrid-Hill estimator- Chang Xu (King's College London)
Abstract: When analysing extreme values, two alternative statistical approaches have historically been held in contention: the block maxima method (or annual maxima method, spurred by hydrological applications) and the peaks-over-threshold. Clamoured amongst statisticians as wasteful of potentially informative data, the block maxima method gradually fell into disfavour whilst peaks-over-threshold-based methodologies climbed to the centre stage of extreme value statistics. In this talk, I will put forward a hybrid method which reconciles these two hitherto disconnected approaches. Appealing in its simplicity, our main result introduces a new universality class of extreme value distributions that discards the customary requirement of a sufficiently large block size for the plausible block maxima-fit to an extreme value distribution. Natural extensions to dependent and/or non-stationary settings are mapped out. In this talk, I will go through a collection of simulation examples across models with different second-order behaviour to show, case by case, how the hybrid-Hill estimator’s finite-sample performance aligns with our asymptotic theory in each setting, thereby providing empirical support for our main theoretical conclusions. Building on these results, a reduced-bias off-shoot is proposed, with the simulations also illustrating the practical gains it delivers. This reduced-bias off-shoot provably outclasses the incumbent maximum likelihood estimation that relies on a numerical fit to the entire sample of block maxima.
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