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 k th nearest neighbor and where scores are allowed to propagate through a random network of points depending on their locations.

Based on their earlier studies of the arcsine law, Pitman and Yor in [1] constructed a widely accepted PD(α, θ) family of random mass-partitions with parameters α∈[0,1),θ+α>0. We propose an alternative model based on generalized perpetuities, which extends the PD family in a continuous manner, incorporating any α ≥ 0. Such a perspective yields a new, concise proof for the stick-breaking (or residual allocation) representations of PD distributions, recovering the classical results of McCloskey and Perman in particular. We further apply this framework to provide a constructive and intuitive proof of Pitman and Yor’s generalized arcsine law concerning the partitions arising from α-stable subordinators for α ∈ (0, 1). The result shows that the random partitions generated by stable subordinators have identical distributions when observed over temporal or spatial intervals. This theorem has a number of significant implications for excursion theory. As a corollary, using purely probabilistic arguments, we obtain general arcsine laws for excursions of d-dimensional Bessel process for 0 < d < 2, and Brownian motion in particular.

Since its introduction by J. Karamata, regular variation has evolved from a purely mathematical concept into a cornerstone of theoretical probability and data analysis. It is extensively studied and applied in different areas. Its significance lies in characterising large deviations, determining the limits of partial sums, and predicting the long-term behaviour of extreme values in stochastic processes. Motivated by various applications, the framework of regular variation has expanded over time to incorporate random observations in more general spaces, including Banach spaces and Polish spaces. In this monograph, we identify three fundamental components of regular variation: scaling, boundedness, and the topology of the underlying space. We explore the role of each component in detail and extend a number of previously obtained results to general topological spaces. Our more abstract approach unifies various concepts appearing in the literature, streamlines existing proofs and paves the way for novel contributions, such as: a generalised theory of (hidden) regular variation for random measures and sets; an innovative treatment of regularly varying random functions and elements scaled by independent random quantities and numerous other advancements. Throughout the text, key results and definitions are illustrated by instructive examples, including extensions of several established models from the literature. By bridging abstraction with practicality, this work aims to deepen both theoretical understanding and methodological applicability of regular variation.

We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence (xi(i) , X-i)(i >= 0), where xi(0) denotes a finite point process on R and X0 denotes a nonnegative random variable of finite mean, we consider the renewal sequence T-i = X-0 + + X-i ,(i >= 0), and corresponding renewal cluster process xi() = Sigma(i >= 0) xi(i)( - T-i). Under mild assumptions on the distribution of (xi, X), we show by coupling methods that the generalized versions of Blackwell's renewal theorem, key renewal theorem, extended renewal theorem and elementary renewal theorem still hold, even with dependence between xi(i) 's and X-i 's.

AbstractThis 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 propose and analyze a permutation test of the tail dependence between two random variables whose marginal distributions are assumed to be known. Justifying the test, we show that the proposed test statistics and their permutation distribution converge to the normal distribution. The analysis is motivated by the recent results of DiCiccio and Romano (J Am Stat Assoc 112(519):1211–1220, 2017) on permutation tests for correlation between random variables. We also provide a simulation study of the size and power properties of the test and an application to financial data.

We study the all-time supremum of the perturbed branching random walk, known to be the endogenous solution to the high-order Lindley equation: W=DmaxY,max1≤i≤N(Wi+Xi),where the {Wi} are independent copies of W, independent of the random vector (Y,N,{Xi}) taking values in R×N×R∞. Under Kesten assumptions, this solution satisfies P(W>t)∼He−αt,t→∞,where α>0 solves the Cramér–Lundberg equation E∑i=1NeαXi=1. This paper establishes the tail asymptotics of W by using the forward iterations of the map defining the fixed-point equation combined with a change of measure along a randomly chosen path. This new approach provides an explicit representation of the constant H and gives rise to unbiased and strongly efficient estimators for the rare event probabilities P(W>t).

We investigate branching processes with immigration in a random environment. Using Goldie’s implicit renewal theory we prove that under a generalized Cramér condition the stationary distribution of such processes has a power law tail. We further show how several methods familiar in the extreme value theory provide a natural and elegant path to their mathematical analysis. In particular, we rely on the point processes theory and the concept of tail process to determine the limiting distribution for the corresponding extremes and partial sums. Since Kesten, Kozlov and Spitzer seminal 1975 paper, it is known that one class of these processes has a close relation with random walks in a random environment. Even in that well studied context, the method we follow yields new results. For instance, we are able to i) move away from the conditions used by Kesten et al., ii) provide precise form of the limiting distribution in their main theorem, and iii) characterize the long term behavior of the worst traps a random walk in random environment encounters when drifting away from the origin.

We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter $0<\alpha<4$. Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.

We describe the asymptotic behavior of the conditional least squares estimator of the offspring mean for subcritical strongly stationary Galton–Watson processes with regularly varying immigration with tail index α∈(1,2). The limit law is the ratio of two dependent stable random variables with indices α∕2 and 2α∕3, respectively, and it has a continuously differentiable density function. We use point process technique in the proofs.

The article determines the asymptotic shape of the extremal clusters in stationary regularly varying random fields. To deduce this result, we present a general framework for the Poisson approximation of point processes on Polish spaces which appears to be of independent interest. We further introduce a novel and convenient concept of anchoring of the extremal clusters for regularly varying sequences and fields. Together with the Poissonian approximation theory, this allows for a concise description of the limiting behavior of random fields in this setting. We apply this theory to shed entirely new light on the classical problem of evaluating local alignments of biological sequences.

We consider the distributional fixed-point equation: R 𝒟= Q ∨( ⋁_i=1^N C_i R_i ), where the {R_i} are i.i.d. copies of R, independent of the vector (Q, N, {C_i}), where N ∈ℕ, Q, {C_i}≥ 0 and P(Q > 0) > 0. By setting W = log R, X_i = log C_i, Y = log Q it is equivalent to the high-order Lindley equation W 𝒟=max{ Y, max_1 ≤ i ≤ N (X_i + W_i) }. It is known that under Kesten assumptions, P(W > t) ∼ H e^-α t, t →∞, where α>0 solves the Cramér-Lundberg equation E [ ∑_j=1^N C_i ^α] = E[ ∑_i=1^N e^α X_i] = 1. The main goal of this paper is to provide an explicit representation for P(W > t), which can be directly connected to the underlying weighted branching process where W is constructed and that can be used to construct unbiased and strongly efficient estimators for all t. Furthermore, we show how this new representation can be directly analyzed using Alsmeyer's Markov renewal theorem, yielding an alternative representation for the constant H. We provide numerical examples illustrating the use of this new algorithm.

We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology.

We study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence arrival rate of the future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or alternatively tends in distribution to an infinite variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes processes as our key example.

The article presents the theory of stationary regularly varying random fields. In this context, we prove a new compound Poisson approximation theorem under appropriate dependence conditions, and demonstrate a couple of effective methods for checking its conditions. The theory is based on an extension of some standard point processes limiting theory to general Polish spaces. Finally, we apply these theoretical results to shed entirely new light on the classical problem of evaluating local alignments of biological sequences.

We propose a notion of convergence of measures with intention of generalizing and unifying several frequently used types of vague convergence. We explain that by general theory of boundedness due to Hu [Hu66], in Polish spaces, this notion of convergence can be always formulated as follows: μn v −→ μ if ∫ fdμn → ∫ fdμ for all continuous bounded functions f with support bounded in some suitably chosen metric. This connects all the related types of vague convergence with the framework of Daley and Vere-Jones [DVJ03] and Kallenberg [Kal17]. In the rest of the note we discuss the vague topology and sufficient conditions for the corresponding notion of convergence in distribution, complementing the theory developed in those two references.

We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of point process convergence theorem. It is designed to preserve the entire information about the temporal ordering of observations which is typically lost in the limit after time scaling. By going beyond the existing asymptotic theory, we are able to prove a new functional limit theorem. Its assumptions are satisfied by a wide class of applied time series models, for which standard limiting theory in the space D of càdlàg functions does not apply. To describe the limit of partial sums in this more general setting, we use the space E of so-called decorated càdlàg functions. We also study the running maximum of partial sums for which a corresponding functional theorem can be still expressed in the familiar setting of space D. We further apply our method to analyze record times in a sequence of dependent stationary observations, even when their marginal distribution is not necessarily regularly varying. Under certain restrictions on dependence among the observations, we show that the record times after scaling converge to a relatively simple compound scale invariant Poisson process.

For a class of stationary regularly varying and weakly dependent multivariate time series (X n ), we prove the so-called complete convergence result for the space–time point processes of the form \(N_{n} = \sum _{i=1}^{n} \delta _{(i/n, \boldsymbol {X}_{i}/a_{n})}.\) As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.