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In this paper, we investigate the asymptotic behavior of supercritical branching Markov processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are L\'evy processes with regularly varying tails. Recently, Ren et al. [Appl. Probab. 61…
We consider sequences $(X_t^N)_{t\geq0}$ of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation of the form $\dot{x}_t=b(x_t)$, where $b(x)={\pmatrix{-\mu 0 0 \lambda}}x+\tau(x)$…
Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several representations of max-stable random fields have been proposed in the literature. One such representation is based on a limit of…
A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes…
This paper investigates extreme value theory for processes obtained by applying transformations to stationary Gaussian processes, also called subordinated Gaussian processes. The main contributions are as follows. First, we refine the…
We consider a one-dimensional stationary time series of fixed duration $T$. We investigate the time $t_{\rm m}$ at which the process reaches the global maximum within the time interval $[0,T]$. By using a path-decomposition technique, we…
Let $M_n$ be the maximum of $n$ zero-mean gaussian variables $X_1,..,X_n$ with covariance matrix of minimum eigenvalue $\lambda$ and maximum eigenvalue $\Lambda$. Then, for $n \ge 70$, $$\Pr\{M_n \ge \lambda \left (2 \log n - 2.5 - \log(2…
The stable-regenerative multiple-stable model has been shown recently to have distinct candidate extremal index and extremal index. To understand further this rare phenomenon, two more results are established here for the double-stable…
We derive functional convergence of the partial maxima stochastic processes of multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. The convergence takes place in the space of…
For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process…
The paper deals with the expected maxima of continuous Gaussian processes $X = (X_t)_{t\ge 0}$ that are H\"older continuous in $L_2$-norm and/or satisfy the opposite inequality for the $L_2$-norms of their increments. Examples of such…
Max-stable processes are widely used to model spatial extremes. These processes exhibit asymptotic dependence meaning that the large values of the process can occur simultaneously over space. Recently, inverted max-stable processes have…
For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index $\alpha \in (0,2)$ and weak dependence conditions. The…
We consider the random field M(t)=\sup_{n\geq 1}\big\{-\log A_{n}+X_{n}(t)\big\}\,,\qquad t\in T\, for a set $T\subset \mathbb{R}^{m}$, where $(X_{n})$ is an iid sequence of centered Gaussian random fields on $T$ and $0<A_{1}<A_{2}<\cdots $…
We establish central limit theorems for a large class of supercritical branching Markov processes in infinite dimension with spatially dependent and non-necessarily local branching mechanisms. This result relies on a fourth moment…
Consider a continuous time Markov chain with rates Q in the state space \Lambda\cup\{0\} with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in \Lambda with rates Q until one of them…
In this paper, we study the asymptotic relation between the maximum of acontinuous order statistics process formed by stationary Gaussian processesand the maximum of this process sampled at discrete time points. It is shown that, these two…
The analysis of spatial extremes requires the joint modeling of a spatial process at a large number of stations and max-stable processes have been developed as a class of stochastic processes suitable for studying spatial extremes. Spatial…
An infinite system of point particles placed in $\mathds{R}^d$ is studied. Its constituents perform random jumps with mutual repulsion described by a translation-invariant jump kernel and interaction potential, respectively. The pure states…
The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs…