arXiv subjects

Krzysztof Dȩbicki

Publications and source records attributed to Krzysztof Dȩbicki.

27 records · Page 2Linked to original sources

arXiv2016

Extremes of a class of nonhomogeneous Gaussian random fields

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 attains its maximum on a segment on $\mathbf{E}$. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.

math.PR↗

arXiv2015

Extremes of stationary Gasussian storage models

For the stationary storage process $\{Q(t), t\ge0\}$, with $ Q(t)=\sup_{ s \ge t}\left(X(s)-X(t)-c(s-t)^β\right), $ where $\{X(t),t\ge 0\}$ is a centered Gaussian process with stationary increments, $c>0$ and $β>0$ is chosen such that $Q(t)$ is finite a.s., we derive exact asymptotics of $\mathbb{P}\left(\sup_{t\in [0,T_u]} Q(t)>u\right)$ and $\mathbb{P}\left(\inf_{t\in [0,T_u]} Q(t)>u\right)$, as $u\rightarrow\infty$. As a by-product we find conditions under which strong Piterbarg property holds.

math.PR↗

arXiv2015

Extremes of vector-valued Gaussian processes: exact asymptotics

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$ as $u\to\infty$, for both locally stationary $X_i$'s and $X_i$'s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.

math.PR↗

arXiv2014

Random Scaling of Gumbel Risks

In this paper we consider the product of two positive independent risks $Y_1$ and $Y_2$. If $Y_1$ is bounded and $Y_2$ has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity, then we show that $Y_1Y_2$ has also distribution in the Gumbel max-domain of attraction. Additionally, if both $Y_1,Y_2$ have log-Weibullian or Weibullian tail behavior, we show that $Y_1Y_2$ has log-Weibullian or Weibullian asymptotic tail behavior, respectively. We present two applications of our results.

math.PR↗

arXiv2014

Parisian Ruin of Self-similar Gaussian Risk Processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

math.PR↗

arXiv2014

On the tail asymptotics of the area swept under the Brownian storage graph

In this paper, the area swept under the workload graph is analyzed: with $\{Q(t) : t\ge0\}$ denoting the stationary workload process, the asymptotic behavior of \[π_{T(u)}(u):={\mathbb{P}}\biggl(\int_0^ {T(u)}Q(r)\,\mathrm{d}r>u\biggr)\] is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of $π_{T(u)}(u)$ are given for the case that $T(u)$ grows slower than $\sqrt{u}$, and then logarithmic asymptotics for (i) $T(u)=T\sqrt{u}$ (relying on sample-path large deviations), and (ii) $\sqrt{u}=\mathrm{o}(T(u))$ but $T(u)=\mathrm{o}(u)$. Finally, the Laplace transform of the residual busy period are given in terms of the Airy function.

math.ST↗

arXiv2014

On the Probability of Conjunctions of Stationary Gaussian Processes

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants $u,T$, define the set of conjunctions $C_{[0,T],u}:=\{t\in [0,T]: \min_{1 \le i \le n} X_i(t) \ge u\}.$ Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of $ P(C_{[0,T],u} \not=φ)$ as $u\to\infty$. Moreover, we establish the Berman sojourn limit theorem for the random process $\{\min_{1 \le i \le n} X_i(t), t\ge0\}$ and derive the tail asymptotics of the supremum of each order statistics process.

math.PR↗

arXiv2013

Tail Asymptotics of Supremum of Certain Gaussian Processes over Threshold Dependent Random Intervals

Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $γ$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^γ]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an independent of $X$ non-negative random variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes.

math.PR↗

arXiv2011

Asymptotics of supremum distribution of a Gaussian process over a Weibullian time

Let $\{X(t):t\in[0,\infty)\}$ be a centered Gaussian process with stationary increments and variance function $σ^2_X(t)$. We study the exact asymptotics of ${\mathbb{P}}(\sup_{t\in[0,T]}X(t)>u)$ as $u\to\infty$, where $T$ is an independent of $\{X(t)\}$ non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion.

math.PR↗