Conference Agenda

In multivariate extreme value analysis, estimating the extremal dependence structure is a challenging task, especially in the context of high-dimensional data. Therefore, a common approach is to reduce the dimensionality by considering only the directions in which extreme values occur. Typically, the underlying models are assumed to be multivariate regularly varying, which under mild assumptions is equivalent to sparse regularly varying, recently introduced by Meyer and Wintenberger (2021). Sparse regular variation has the advantage of capturing the sparsity structure in which extreme events occur better than multivariate regular variation. Therefore, in this talk, we use the concept of sparse regular variation to present different information criteria for the number of directions in which extreme events occur, such as a Bayesian information criterion (BIC), a mean-squared error information criterion (MSEIC) and a quasi-Akaike information criterion (QAIC) based on the Gaussian likelihood function. A result is that the AIC of Meyer and Wintenberger (2023) and the MSEIC are inconsistent information criteria whereas the BIC and the QAIC are consistent information criteria. Finally, the performance of the different information criteria is compared in a simulation study.

Principal component analysis (PCA) is one of the most popular dimension reduction techniques in statistics and is especially powerful when a multivariate distribution is concentrated near a lower-dimensional subspace. Multivariate extreme value distributions have turned out to provide challenges for the application of PCA since their constraint support impedes the detection of lower-dimensional structures and heavy-tails can imply that second moments do not exist, thereby preventing the application of classical variance-based techniques for PCA. We adapt PCA to max-stable distributions using a regression setting and employ max-linear maps to project the random vector to a lower-dimensional space while preserving max-stability. We also provide a characterization of those distributions which allow for a perfect reconstruction from the lower-dimensional representation. Finally, we demonstrate how an optimal projection matrix can be consistently estimated and show viability in practice with a simulation study and application to a benchmark dataset.

A common feature observed in large real-world networks are degree distributions that resemble power laws. Since this has serious practical implications, many models have been proposed over time that aim to reflect this property. One of these is the class of preferential attachment models, which gained popularity soon after their introduction by Barabási and Albert in 1999. They describe a discrete-time growing graph process in which, at each time step, a newly added vertex randomly establishes a certain number of edges to existing vertices with a probability that is an affine function of their degrees. This 'rich-get-richer' dynamic provides an intuitive explanation for power-law distributions and has furthermore been proven to lead to these distributions asymptotically.

In this talk, we provide a complementary approach aimed at analysing individual vertices and their interactions in large networks. To this end, we select a fixed number of the oldest vertices and let them evolve for a heavy-tailed random number of time steps. Utilising tools from extreme value theory, such as the tail coefficient and spectral measure, we can then make predictions for the chosen degree vector in large networks, which we interpret as just the extremal realisations of our model. We discuss several model specifications, such as finite versus infinite dimensions, and fixed versus random numbers of outgoing edges per vertex.

In this presentation, we give explicit bounds on the Wasserstein and the Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard Normal distribution, using the results proved by Last, Peccati and Schulte. Relying on the theory developed in the work of Saulis and Statulevicius and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities and Normal approximation bounds with Cramér correction terms for the same variables. The aforementioned results are then applied to Poisson shot-noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models which generalizes classical Hawkes processes). This extends the recent results available in the literature regarding the Normal approximation and moderate deviations.