Conference Agenda

We investigate limit theorems for extreme values of the classical inversion and descent statistics on symmetric groups and related permutation groups. The central limit theorem (CLT) is known to hold for these permutation statistics. We can achieve comparable results for their maxima in a suitable triangular array $(X_{n1}, \ldots, X_{nk_n})$. The triangular array is required as these discretely distributed statistics have a degenerate extreme value behavior otherwise. It is important to discuss the regime of the row lengths $k_n$ since an overly large triangular array is still degenerate. We demonstrate the use of large deviations theory and high-dimensional Gaussian approximation in order to transfer asymptotic normality to the extreme values, and we discuss general findings for extreme values in triangular arrays.

In this work, we aim to model spatio-temporal heavy precipitation events, which enhances the understanding of their causes and their prediction. Similar to Koh et al. (2023), we propose to use spatio-temporal point processes with covariates. However, we model the intensity measure of the spatio-temporal point process differently with a similar approach as Baddeley et al. (2012) who estimate the intensity measure nonparametrically. This point process model can be extended by adding marks e.g. the spatial extent (see Oesting & Huser, 2022) and clustering to obtain a clustered marked point process model.

References:

A. Baddeley, Y.-M. Chang, Y. Song, and R. Turner. Nonparametric estimation of the dependence of a spatial point process on spatial covariates. Statistics and its interface, 5(2):221-236, 2012.

J. Koh, F.Pimont, J.-L. Dupuy, and T. Opitz. Spatiotemporal wildfire modeling through point processes with moderate and extreme mark. The Annals of Applied Statistics, 17(1):560-582, 2023.

M. Oesting and R. Huser. Patterns in spatio-temporal extremes. arXiv preprint arXiv:2212.11001, 2022.

Consider a random walk $S_k^{(d)}$, $k\geq 0$, in $d$-dimensional Euclidean space with square integrable centred increments such that the expected square norm of the increment is one. The values of this random walk for $k=0,\dots,n$ normalised by $\sqrt{n}$ are considered as a finite metric space $\mathcal{Z}_n^{(d)}$ which is embedded in $\mathbb{R}^d$ with the induced metric. Under condition of uniform smallness type on the $d$ components of the incement, Kabluchko and Marynych (2022) proved that, as $n$ and $d$ go to infinity in any regime, then the metric space $\mathcal{Z}_n^{(d)}$ converges in probability in the Gromov--Hausdorff metric to the Wiener spiral. The latter space is the space of all indicators $\mathbf{1}_{[0,t]}$, $t\in[0,1]$, embedded in $L^2([0,1])$, equivalently, the interval $[0,1]$ with the metric $r(t,s)=\sqrt{|t-s|}$.

In their subsequent paper, Kabluchko and Marynych (2023) showed that in the heavy-tailed case with $\alpha\in(0,1)$, the limiting metric space is random and is derived from an infinite-dimensional version of a subordinator (called a crinckled subordinator), assuming certain condition on the joint growth regime of $n$ and $d$.

We study an extremal version of this setting, where the random walk is replaced by a sequence of successive maxima and the underlying metric in $\mathbb{R}^d$ is taken to be $\ell_\infty$. We show that the limiting metric space is the extremal version of the crinckled subordinator which is derived from a Poisson process $\{x_k, y_k \} $ on $[0,1] \times \mathbb{R}_{+}$ with intensity measure being the product of the Lebesgue meausure and the tail measure derived from the distribution of the components. The limit is isometric to $[0, 1]$ with the metric $r(s, t)$ given by the maximum of $y_k$ over all $x_k$ with $x_k \in [s, t]$.

Joint work with Ilya Molchanov (Bern).

In order to describe the extremal behaviour of some stochastic process $X$ approaches from univariate extreme value theory are typically generalized to the spacial domain. Besides max-stable processes, that can be used in analogy to the block maxima approach, a generalized peaks-over-threshold approach can be used, allowing us to consider single extreme events. These can be flexibly defined as exceedances of a risk functional $\ell$, such as a spatial average, applied to $X$. Inference for the resulting limit process, the so-called $\ell$-Pareto process, requires the evaluation of $\ell(X)$ and thus the knowledge of the whole process $X$. In practical application we face the challenge that observations of $X$ are only available at single sites. To overcome this issue, we propose a two-step MCMC-algorithm in a Bayesian framework. In a first step, we sample from $X$ conditionally on the observations in order to evaluate which observations lead to $\ell$-exceedances. In a second step, we use these exceedances to sample from the posterior distribution of the parameters of the limiting $\ell$-Pareto process. Alternating these steps results in a full Bayesian model for the extremes of $X$. We show that, under appropriate assumptions, the probability of classifying an observation as $\ell$-exceedance in the first step converges to the desired probability. Furthermore, given the first step, the distribution of the Markov chain constructed in the second step converges to the posterior distribution of interest. Our procedure is compared to the Bayesian version of the standard procedure in a simulation study.