Related papers: Stationary max-stable fields associated to negativ…

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. For statistical inference it is often assumed that…

In this short note we will provide a sufficient and necessary condition to have uniqueness of the location of the maximum of a stochastic process over an interval. The result will also express the mean value of the location in terms of the…

Max-stable random fields can be constructed according to Schlather (2002) with a random function or a stationary process and a kind of random event magnitude. These are applied for the modelling of natural hazards. We simply extend these…

This paper studies the winding of a continuously differentiable Gaussian stationary process $f:\mathbb{R}\to\mathbb{C}$ in the interval $[0,T]$. We give formulae for the mean and the variance of this random variable. The variance is shown…

The seminal papers of Pickands [1,2] paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including…

Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion. As an application of a recent result of ours on exponential functionals of Brownian motion, we show in this paper that, for every fixed $t>0$, the process given by…

Let $\{X(t), t\geq0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004), which we refer to as Piterbarg's max-discretisation theorem gives the joint asymptotic behaviour ($T\to…

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…

We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$…

Let $\{X_{n}(t), t\in[0,\infty)\}, n\in\mathbb{N}$ be a sequence of centered dependent stationary Gaussian processes. The limit distribution of $\sup_{t\in[0,T(n)]}|X_{n}(t)|$ is established as $r_{n}(t)$, the correlation function of…

Employing the optimal fluctuation method (OFM), we study the large deviation function of long-time averages $(1/T)\int_{-T/2}^{T/2} x^n(t) dt$, $n=1,2, \dots$, of centered stationary Gaussian processes. These processes are correlated and,…

Let $X_H(t), t\ge 0$ be a fractional Brownian motion with Hurst index $H\in(0,1}$ and define a gamma-reflected process $W_\Ga(t)=X_H(t)-ct-\gammainf_{s\in[0,t]}\left(X_H(s)-cs \right)$, $t\ge0$ with $c>0,\gamma \in [0,1]$ two given…

Let $\{X(t):t\in[0,\infty)\}$ be a centered Gaussian process with stationary increments and variance function $\sigma^2_X(t)$. We study the exact asymptotics of ${\mathbb{P}}(\sup_{t\in[0,T]}X(t)>u)$ as $u\to\infty$, where $T$ is an…

Let $a$ be a finite signed measure on $[-r, 0]$ with $r \in (0, \infty)$. Consider a stochastic process $(X^{(\vartheta)}(t))_{t\in[-r,\infty)}$ given by a linear stochastic delay differential equation \[ \mathrm{d} X^{(\vartheta)}(t) =…

Brownian motion is the only random process which is Gaussian, stationary and Markovian. Dropping the Markovian property, i.e. allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst…

We consider a process $Z$ on the real line composed from a L\'evy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$…

We study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a L\'evy process $\mathbf{Y}$ by partial resetting, that is the position of the process $\mathbf{X}$ at a Poisson moment equals $c$ times its position right before…

In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. Unlike previously known facts in this field, our main…

The process $(G_t)_{t\in[0,T]}$ is referred to as a fractional Gaussian process if the first-order partial derivative of the difference between its covariance function and that of the fractional Brownian motion $(B^H_t)_{t\in[0,T ]}$ is a…

Let $G=\{G(x),x\ge 0\}$ be a mean zero Gaussian process with stationary increments and set $\sigma^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(\eta)<\ff$, where $\eta=N(0,1)$. When $\sigma^2$ is regularly varying at zero…