arXiv subjects

Krzysztof Dębicki

Publications and source records attributed to Krzysztof Dębicki.

10 recordsLinked to original sources

arXiv2026

Sojourns of Vector-Valued Stationary Gaussian Random Fields

For a centered, homogeneous R^d-valued Gaussian random field X(t), t in R^k, with covariance matrix function R(s,t) = E[X(s) X(t)^T], we investigate the exact asymptotics of kappa_u(x) = P( theta(u) * integral over [0,T]^k of 1{X(t) > u b} dt > x ), where b = (b_1, ..., b_d)^T, as u -> infinity, with x >= 0 and T > 0, and theta(u) is a scaling function related to the expansion of R(s,t) around (0,0). We extend Berman's approach and the uniform double-sum method to this vector-valued random-field setting. Our formulation uses weaker assumptions and permits varying domains T=T_u at an admissible rate. It also yields exact tail asymptotics for the supremum functional of X.

math.PR↗

arXiv2026

Distributional and Extremal Behaviour of Brownian Motion with Exponential Resetting

We study the distributional and asymptotic properties of the supremum of Brownian motion with drift and exponential resetting. We obtain an explicit renewal-type formula for the distribution of the supremum and then derive an approximation for its survival function. Moreover, we find the asymptotics of the tail distribution of the supremum and infimum. We also consider the stationary case and give a new explicit expression for the two-dimensional distributions of such processes.

math.PR↗

arXiv2022

On Berman functions

For fractional Brownian motion with Hurst parameter H the Berman constant is defined. In this paper we consider a general random field (rf) Z that is a spectral rf of some stationary max-stable rf X and derive the properties of the corresponding Berman functions. In particular, we show that Berman functions can be approximated by the corresponding discrete ones and derive interesting representations of those functions which are of interest for Monte Carlo simulations, which are presented in this article.

math.PR↗

arXiv2021

Derivatives of sup-functionals of fractional Brownian motion evaluated at H=1/2

We consider a family of sup-functionals of (drifted) fractional Brownian motion with Hurst parameter $H\in(0,1)$. This family includes, but is not limited to: expected value of the supremum, expected workload, Wills functional, and Piterbarg-Pickands constant. Explicit formulas for the derivatives of these functionals as functions of Hurst parameter evaluated at $H=\tfrac{1}{2}$ are established. In order to derive these formulas, we develop the concept of derivatives of fractional $α$-stable fields introduced by Stoev \& Taqqu (2004) and propose Paley-Wiener-Zygmund representation of fractional Brownian motion.

math.PR↗

arXiv2021

Bounds for expected supremum of fractional Brownian motion with drift

We provide upper and lower bounds for the mean ${\mathscr M}(H)$ of $\sup_{t\geqslant 0} \{B_H(t) - t\}$, with $B_H(\cdot)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$. We find bounds in (semi-)closed-form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$, where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$, the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$, $H\in(0,\tfrac{1}{2}]$, which is tight around $H=\tfrac{1}{2}$.

math.PR↗

arXiv2017

On Extremal Index of max-stable stationary processes

In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary process $W(t),t\in R$ as $$\mathcal{H}_W^δ= \lim_{T\to\infty} T^{-1} E{ \left(\sup_{t\in δZ \cap [0,T]} e^{W(t)}\right) },\ δ\ge 0$$ (set $0 Z=R$ if $δ=0$) and the extremal index of the associated max-stable stationary process $ξ_W$. We derive several new formulas and obtain lower bounds for $\mathcal{H}_W^δ$ if $W$ is a Gaussian or a Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.

math.PR↗

arXiv2016

Generalized Pickands constants and stationary max-stable processes

Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

math.PR↗

arXiv2015

Extremes of multidimensional Gaussian processes

This paper considers extreme values attained by a centered, multidimensional Gaussian process $X(t)= (X_1(t),\ldots,X_n(t))$ minus drift $d(t)=(d_1(t),\ldots,d_n(t))$, on an arbitrary set $T$. Under mild regularity conditions, we establish the asymptotics of \[\log\mathbb P\left(\exists{t\in T}:\bigcap_{i=1}^n\left\{X_i(t)-d_i(t)>q_iu\right\}\right),\] for positive thresholds $q_i>0$, $i=1,\ldots,n$, and $u\to\infty$. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.

math.PR↗

arXiv2014

On the infimum attained by the reflected fractional Brownian motion

Let $\{B_H(t):t\ge 0\}$ be a fractional Brownian motion with Hurst parameter $H\in(\frac{1}{2},1)$. For the storage process $Q_{B_H}(t)=\sup_{-\infty\le s\le t} \left(B_H(t)-B_H(s)-c(t-s)\right)$ we show that, for any $T(u)>0$ such that $T(u)=o(u^\frac{2H-1}{H})$, \[\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_H}(s)>u)\sim\mathbb P(Q_{B_H}(0)>u),\quad\text{as}\quad u\to\infty.\] This finding, known in the literature as the strong Piterbarg property, is in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.

math.PR↗

arXiv2013

Extremes of homogeneous Gaussian random fields

Let $\{X(s,t):s,t\geqslant 0\}$ be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function $r(s,t)=Cov(X(s,t),X(0,0))$ such that \[r(s,t)=1-|s|^{α_1}-|t|^{α_2}+o(|s|^{α_1}+|t|^{α_2}), \quad s,t \to 0,\] with $α_1,α_2\in(0,2],$ and $r(s,t)<1$ for $(s,t)\neq(0,0)$. In this contribution we derive an exact asymptotic expansion (as $u\to \infty$) of $$\mathbb{P}\left(\sup_{(s n_1(u),t n_2(u))\in\left[0,x\right]\times\left[0,y\right]}X(s,t)\leqslant u\right),$$ where $n_1(u)n_2(u)=u^{2/α_1+2/α_2}Ψ(u)$, which holds uniformly for $(x,y) \in [ A , B ]^2$ with $ A , B $ two positive constants and $Ψ$ the survival function of an $N(0,1)$ random variable. We apply our findings to the analysis of asymptotics of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally we determine the extremal index of the discretised random field determined by $X(s,t)$.

math.PR↗