We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizontal displacement as well as renewal-process arguments.

We consider stationary configurations of points in Euclidean space that are marked by positive random variables called scores. The scores are allowed to depend on the relative positions of other points and outside sources of randomness. Such models have been thoroughly studied in stochastic geometry, e.g. in the context of random tessellations or random geometric graphs. It turns out that in a neighborhood of a point with an extreme score it is possible to rescale positions and scores of nearby points to obtain a limiting point process, which we call the tail configuration. Under some assumptions on dependence between scores, this local limit determines the global asymptotics for extreme scores within increasing windows in $\mathbb{R}^d$. The main result establishes the convergence of rescaled positions and clusters of high scores to a Poisson cluster process, quantifying the idea of the Poisson clumping heuristic by Aldous (1989, in the point process setting). In contrast to the existing results, our framework allows for explicit calculation of essentially all extremal quantities related to the limiting behavior of extremes. We apply our results to models based on (marked) Poisson processes where the scores depend on the distance to the kth nearest neighbor and where scores are allowed to propagate through a random network of points depending on their locations.

We use the framework of multivariate regular variation to analyse the extremal behaviour of preferential attachment models. To this end, we follow a directed linear preferential attachment model for a random, heavy-tailed number of steps in time and treat the incoming edge count of all existing nodes as a random vector of random length. By combining martingale properties, moment bounds and a Breiman type theorem we show that the resulting quantity is multivariate regularly varying, both as a vector of fixed length formed by the edge counts of a finite number of oldest nodes, and also as a vector of random length viewed in sequence space. A Pólya urn representation allows us to explicitly describe the extremal dependence between the degrees with the help of Dirichlet distributions. As a by-product of our analysis we establish new results for almost sure convergence of the edge counts in sequence space as the number of nodes goes to infinity.

Considering a double-indexed array $(Y_{n,i:\,n\ge 1,i\ge 1})$ of non-negative regularly varying random variables, we study the random-length weighted sums and maxima from its ‘row’ sequences. These sums and maxima may have the same tail and extremal indices (Markovich and Rodionov 2020). The main constraints of the latter results are that there exists a unique series in a scheme of series with the minimum tail index and the tail of the term number is lighter than the tail of the terms. Here, a bounded random number of series are allowed to have the minimum tail index and the tail of the term number may be heavier than the tail of the terms. We derive the tail and extremal indices of the weighted non-stationary random-length sequences under a broader set of conditions than in Markovich and Rodionov (2020). We provide examples of random sequences for which the assumptions are valid. Perspectives in adopting the results in different application areas are formulated.

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C$\beta $E ensemble. We prove convergence in distribution of the centered maxima (of the real and imaginary parts) toward the sum of a Gumbel variable and another independent variable, which we characterize as the total mass of a ‘derivative martingale’. We also provide a description of the landscape near extrema points.

We derive some key extremal features for stationary kth-order Markov chains that can be used to understand how the process moves between an extreme state and the body of the process. The chains are studied given that there is an exceedance of a threshold, as the threshold tends to the upper endpoint of the distribution. Unlike previous studies with $k>1$, we consider processes where standard limit theory describes each extreme event as a single observation without any information about the transition to and from the body of the distribution. Our work uses different asymptotic theory which results in non-degenerate limit laws for such processes. We study the extremal properties of the initial distribution and the transition probability kernel of the Markov chain under weak assumptions for broad classes of extremal dependence structures that cover both asymptotically dependent and asymptotically independent Markov chains. For chains with $k>1$, the transition of the chain away from the exceedance involves novel functions of the k previous states, in comparison to just the single value, when $k=1$. This leads to an increase in the complexity of determining the form of this class of functions, their properties, and the method of their derivation in applications. We find that it is possible to derive an affine normalization, dependent on the threshold excess, such that non-degenerate limiting behaviour of the process, in the neighbourhood of the threshold excess, is assured for all lags. We find that these normalization functions have an attractive structure that has parallels to the Yule–Walker equations. Furthermore, the limiting process is always linear in the innovations. We illustrate the results with the study of kth-order stationary Markov chains with exponential margins based on widely studied families of copula dependence structures.

In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants $\mathcal{H}^\delta_\alpha$ using a family of estimators $\xi^\delta_\alpha(T)$, $T>0$, where $\alpha\in(0,2]$ is the Hurst parameter, and $\delta\geq0$ is the step size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_\alpha^0 - \mathcal{H}_\alpha^\delta$, whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case $\alpha\in(0,1]$ and agrees up to logarithmic terms for $\alpha\in(1,2)$. Moreover, we show that all moments of $\xi_\alpha^\delta(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^{\alpha}\}$, as T becomes large.

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata’s Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes.

Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution function F, say that the $n^{\rm \scriptsize}$th observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let $p_n(F) \equiv p_{n, d}(F)$ denote the probability that the $n^{\rm \scriptsize}$th observation sets a record. There are many interesting questions to address concerning pn and multivariate records more generally, but this short paper focuses on how pn varies with F, particularly if, under F, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called negative record-setting probability dependence (NRPD) and positive record-setting probability dependence (PRPD), relate these notions to existing notions of dependence, and for fixed $d \geq 2$ and $n \geq 1$ prove that the image of the mapping pn on the domain of NRPD (respectively, PRPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$), where $p^*_n$ is the record-setting probability for any continuous F governing independent coordinates.

Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$. With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as ‘model aggregation’.

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed exponential random variables.

Graphical models with heavy-tailed factors can be used to model extremal dependence or causality between extreme events. In a Bayesian network, variables are recursively defined in terms of their parents according to a directed acyclic graph (DAG). We focus on max-linear graphical models with respect to a special type of graph, which we call a tree of transitive tournaments. The latter is a block graph combining in a tree-like structure a finite number of transitive tournaments, each of which is a DAG in which every two nodes are connected. We study the limit of the joint tails of the max-linear model conditionally on the event that a given variable exceeds a high threshold. Under a suitable condition, the limiting distribution involves the factorization into independent increments along the shortest trail between two variables, thereby imitating the behaviour of a Markov random field.

We are also interested in the identifiability of the model parameters in the case when some variables are latent and only a subvector is observed. It turns out that the parameters are identifiable under a criterion on the nodes carrying the latent variables which is easy and quick to check.

We study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.

Consider a well-shuffled deck of cards of n different types where each type occurs m times. In a complete feedback game, a player is asked to guess the top card from the deck. After each guess, the top card is revealed to the player and is removed from the deck. The total number of correct guesses in a complete feedback game has attracted significant interest in the past few decades. Under different regimes of m, n, the expected number of correct guesses, under the greedy (optimal) strategy, has been obtained by various authors, while there are not many results available about the fluctuations. In this paper we establish a central limit theorem with Berry–Esseen bounds when m is fixed and n is large. Our results extend to the case of decks where different types may have different multiplicity, under suitable assumptions.

Extreme value theory plays an important role in providing approximation results for the extremes of a sequence of independent random variables when their distribution is unknown. An important one is given by the generalised Pareto distribution $H_\gamma(x)$ as an approximation of the distribution $F_t(s(t)x)$ of the excesses over a threshold t, where s(t) is a suitable norming function. We study the rate of convergence of $F_t(s(t)\cdot)$ to $H_\gamma$ in variational and Hellinger distances and translate it into that regarding the Kullback–Leibler divergence between the respective densities.

Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any $t>0,$ the density (with respect to the $(d+1)$-dimensional Lebesgue measure) of the pair $\big(M_t,X_t\big)$ is a weak solution of a Fokker–Planck partial differential equation on the closed set $\big\{(m,x)\in \mathbb{R}^{d+1},\,{m\geq x^1}\big\},$ using an integral expansion of this density.

We consider a branching random walk on a d-ary tree of height n ($n \in \mathbb{N}$), in the presence of a hard wall which restricts each value to be positive, where d is a natural number satisfying $d\geqslant2$. We consider the behaviour of Gaussian processes with long-range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of vertices. We observe a relation with the expected maximum of the processes. We find the probability of the event that the branching random walk is positive at every vertex in the nth generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$.

This 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 distribution of $\max\{X_j\,:\, j=0,\ldots, K(t)\}$, where K(t) is the number of independent and identically distributed observations $(X_j)$ arriving up to the time t according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.

We study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U-statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U-statistics. The latter convergence particularly occurs in the $\mathcal M_0$-topology.

$U{\hbox{-}}\textrm{max}$ statistics were introduced by Lao and Mayer in 2008. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons and polyhedra formed by random points on a circle, ellipse, etc. The main method to study limit theorems for $U{\hbox{-}}\textrm{max}$ statistics is via a Poisson approximation. In this paper we consider a general class of kernels defined on a circle, and we prove a universal limit theorem with the Weibull distribution as a limit. Its parameters depend on the degree of the kernel, the structure of its points of maximum, and the Hessians of the kernel at these points. Almost all limit theorems known so far may be obtained as simple special cases of our general theorem. We also consider several new examples. Moreover, we consider not only the uniform distribution of points but also almost arbitrary distribution on a circle satisfying mild additional conditions.