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We study translation-invariant determinantal random point fields on the real line. We prove, under quite general conditions, that the smallest nearest spacings between the particles in a large interval have Poisson statistics as the length…
In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical "unbiased behavior" with exponential…
The key to successful statistical analysis of bivariate extreme events lies in flexible modelling of the tail dependence relationship between the two variables. In the extreme value theory literature, various techniques are available to…
We study the process of suitably normalized successive return times to rare events in the setting of infinite-measure preserving dynamical systems. Specifically, we consider small neighborhoods of points whose measure tends to zero. We…
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…
Spatial Poisson point processes on finite-dimensional Euclidean space provide fundamental mathematical tools for modeling random spatial point patterns. In this paper, we introduce and analyze several Poisson-type spatial point processes.…
This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that…
Consider that the coordinates of $N$ points are randomly generated along the edges of a $d$-dimensional hypercube (random point problem). The probability that an arbitrary point is the $m$th nearest neighbor to its own $n$th nearest…
We consider the extremal shot noise defined by $$M(y)=\sup\{mh(y-x);(x,m)\in\Phi\},$$ where $\Phi$ is a Poisson point process on $\bbR^d\times (0,+\infty)$ with intensity $\lambda dxG(dm)$ and $h:\bbR^d\to [0,+\infty]$ is a measurable…
Observing a load process above high thresholds, modeling it as a pulse process with random occurrence times and magnitudes, and extrapolating life-time maximum or design loads from the data is a common task in structural reliability…
Whether an extreme observation is an outlier or not, depends strongly on the corresponding tail behaviour of the underlying distribution. We develop an automatic, data-driven method to identify extreme tail behaviour that deviates from the…
Assume that a stochastic processes can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation"…
Define the scaled empirical point process on an independent and identically distributed sequence $\{Y_i: i\le n\}$ as the random point measure with masses at $a_n^{-1} Y_i$. For suitable $a_n$ we obtain the weak limit of these point…
Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations.. We consider a set of $N$ independent and…
This paper studies extremal quantiles under two-way clustered dependence. We show that the limiting distribution of unconditional intermediate-order tail quantiles is Gaussian. This result is notable because two-way clustering typically…
We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting…
In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree,…
Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $\phi(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general…
We consider two independent random variables with the given tail asymptotic (e.g. power or exponential). We find tail asymptotic for their sum and product. This is done by some cumbersome but purely technical computations and requires the…
We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincar\'{e} inequality is suboptimal, a phenomenon known as superconcentration. In this paper, we establish a…