arXiv subjects

Enkelejd Hashorva

Publications and source records attributed to Enkelejd Hashorva.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the continuity of Pickands constants

For a non-negative separable random field $Z(t), t\in \mathbb{R}^d$ satisfying some mild assumptions we show that \begin{eqnarray*} H_Z^δ= \lim_{T\to\infty} \frac{1}{T^d} E \{\sup_{ t\in [0,T]^d \cap δ\mathbb{Z}^d } Z(t) \} <\infty \end{eqnarray*} for $δ\ge 0$ where $0 \mathbb{Z}^d := \mathbb{R}^d$ and prove that $H_Z^0$ can be approximated by $H_Z^δ$ if $δ$ tends to 0. These results extend the classical findings for the Pickands constants $H_{Z}^δ$, defined for $Z(t)= \exp\left( \sqrt{ 2} B_α(t)- |t|^{2α}\right), t\in \mathbb{R}$ with $B_α$ a standard fractional Brownian motion with Hurst parameter $α\in (0,1]$. The continuity of $H_{Z}^δ$ at $δ=0$ is additionally shown for two particular extensions of Pickands constants.

math.PR↗

arXiv2021

Multivariate Max-Stable Processes and Homogeneous Functionals

Multivariate max-stable processes are important for both theoretical investigations and various statistical applications motivated by the fact that these are limiting processes, for instance of stationary multivariate regularly varying time series, [1]. In this contribution we explore the relation between homogeneous functionals and multivariate max-stable processes and discuss the connections between multivariate max-stable process and zonoid / max-zonoid equivalence. We illustrate our results considering Brown-Resnick and Smith processes.

math.PR↗

arXiv2021

Sojourn times of Gaussian related random fields

This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum o f random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes.

math.PR↗

arXiv2020

Finite-time ruin probability for correlated Brownian motions

Let $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $ρ\in (-1,1)$ and define the joint survival probability of both supremum functionals $π_ρ(c_1,c_2; u, v)$ by $$π_ρ(c_1,c_2; u, v)=\mathbb{P}\left(\sup_{s \in [0,1]} \left(W_1(s)-c_1s\right)>u,\sup_{t \in [0,1]} \left(W_2(t)-c_2t\right)>v\right) ,$$ where $c_1,c_2 \in \mathbb{R}$ and $u,v$ are given positive constants. Approximation of $π_ρ(c_1,c_2; u, v) $ is of interest for the analysis of ruin probability in bivariate Brownian risk model as well as in the study of bivariate test statistics. In this contribution we derive tight bounds for $π_ρ(c_1,c_2; u, v)$ in the case $ρ\in (0,1)$ and obtain precise approximations by letting $u\to \infty$ and taking $v= au$ for some fixed positive constant $a$ and $ρ\in (-1,1).$

math.PR↗

arXiv2020

Boundary non-crossing probabilities of Gaussian processes: sharp bounds and asymptotics

We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm P\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by some arbitrary compact separable metric space $\mathbb T$. We obtain both upper and lower bounds for $P_{f,u}$. The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $P_{y f,u}$, $y \to+\infty$, which are two-term approximations (up to $o(y)$). The asymptotics are formulated in terms of the solution $\tilde f$ to the constrained optimization problem $$ \|h\|_{\mathbb H_X}\to \min, \quad h\in \mathbb H_X, h\ge f $$ in the reproducing kernel Hilbert space $\mathbb H_X$ of $X$. Several applications of the results are further presented.

math.PR↗

arXiv2020

Multivariate Extremes Over a Random Number of Observations

The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max-stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable distributions. By means of an inversion method, we derive a semiparametric composite-estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large-sample theory of the composite-estimator and illustrate its finite-sample performance via a simulation study.

stat.ME↗

arXiv2019

Approximation of Supremum of Max-Stable Stationary Processes and Pickands Constants

Let $X(t),t\in \mathbb{R}$ be a stochastically continuous stationary max-stable process with Fréchet marginals $Φ_α, α>0$ and set $M_X(T)=\sup_{t \in [0,T]} X(t),T>0$. In the light of the seminal articles [1,2], it follows that $A_T=M_X(T)/T^{1/α}$ converges in distribution as $T\to \infty$ to $\mathcal{H}_Z^{1/α} X(1)$, where $\mathcal{H}_Z$ is the Pickands constant corresponding to the spectral process $Z$ of $X$. In this contribution we derive explicit formulas for $\mathcal{H}_Z$ in terms of $Z$ and show necessary and sufficient conditions for its positivity. From our analysis it follows that $A_T^β,T>0$ is uniformly integrable for any $β\in (0,α)$. Further, we discuss the dissipative Rosiński (or mixed moving maxima) representation of $X$. Additionally, for Brown-Resnick $X$ we show the validity of the celebrated Slepian inequality and obtain lower bounds on the growth of supremum of Gaussian processes with stationary increments by exploiting the link between Pickands constants and Wills functional. Moreover, we derive upper bounds for supremum of centered Gaussian processes given in terms of Wills functional, and discuss the relation between Pickands and Piterbarg constants.

math.PR↗

arXiv2019

Extremes of Vector-Valued Gaussian Processes

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 Slepian's Lemma, Borell-TIS and Piterbarg inequalities there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting obtaining the exact asymptotics of the exceedance probabilities for both stationary and non-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and the operator fractional Ornstein-Uhlenbeck process.

math.PR↗

arXiv2018

Simultaneous Ruin Probability for Two-Dimensional Brownian and Lévy Risk Models

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite-time horizon. This is not the case for the simultaneous ruin probability in two-dimensional Brownian risk model. Resorting on asymptotic theory, we derive in this contribution approximations of both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity. Given the interest in proportional reinsurance, we consider in some details the case where the correlation is 1. This model is tractable allowing for explicit formulas for the simultaneous ruin probability for linearly dependent spectrally positive Lévy processes. Examples include perturbed Brownian and gamma Lévy processes.

math.PR↗

arXiv2018

Tail measure and tail spectral process of regularly varying time series

The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in Owada and Samorodnitsky (2012) and Basrak and Segers (2009). Our main result is to prove in an abstract framework that there is a one to one correspondance between these two objets, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For non negative time series, we recover results explicitly or implicitly known in the theory of max-stable processes.

math.PR↗

arXiv2018

Domination of Sample Maxima and Related Extremal Dependence Measures

For a given $d$-dimensional distribution function (df) $H$ we introduce the class of dependence measures $ μ(H,Q) = - \mathbb{E}\{ \ln H(Z_1, \ldots, Z_d)\},$ where the random vector $(Z_1, \ldots, Z_d)$ has df $Q$ which has the same marginal df's as $H$. If both $H$ and $Q$ are max-stable df's, we show that for a df $F$ in the max-domain of attraction of $H$, this dependence measure explains the extremal dependence exhibited by $F$. Moreover we prove that $μ(H,Q)$ is the limit of the probability that the maxima of a random sample from $F$ is marginally dominated by some random vector with df in the max-domain of attraction of $Q$. We show a similar result for the complete domination of the sample maxima which leads to another measure of dependence denoted by $λ(Q,H)$. In the literature $λ(H,H)$ with $H$ a max-stable df has been studied in the context of records, multiple maxima, concomitants of order statistics and concurrence probabilities. It turns out that both $μ(H,Q)$ and $λ(Q,H)$ are closely related. If $H$ is max-stable we derive useful representations for both $μ(H,Q)$ and $λ(Q,H)$. Our applications include equivalent conditions for $H$ to be a product df and $F$ to have asymptotically independent components.

math.PR↗

arXiv2017

Approximation of Sojourn Times of Gaussian Processes

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes $X$, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of [1,2] for non-stationary $X$. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.

math.PR↗

arXiv2017

Tail asymptotics of light-tailed Weibull-like sums

We consider sums of $n$ i.i.d. random variables with tails close to $\exp\{-x^β\}$ for some $β>1$. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg & Resnick (1993) are discussed from the point of view of tails rather of densities, using a somewhat different angle, and supplemented with bounds, results on a random number $N$ of terms, and simulation algorithms.

math.PR↗

arXiv2017

Extremes of $γ$-reflected Gaussian process with stationary increments

For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_γ(t)=X(t)-ct-γ\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $γ$-reflected process, where $γ\in (0,1)$. This process is introduced in the context of risk theory to model surplus process that include tax payments of loss-carry forward type.In this contribution we derive asymptotic approximations of both the ruin probability and the joint distribution of first and last passage times given that ruin occurs. We apply our findings to the cases with $X$ being the multiplex fractional Brownian motion and the integrated Gaussian processes. As a by-product we derive an extension of Piterbarg inequality \KD{for} threshold-dependent random fields.

math.PR↗

arXiv2017

Extremal behaviour of hitting a cone by correlated Brownian motion with drift

This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbolμt\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion starting at the point $\mathbf{x}_u=-\boldsymbolαu$ with $\boldsymbolα\in \mathbb{R}^d$, $\boldsymbolμ \in \mathbb{R}^d$ and $\mathcal{U}=\prod_{i=1}^d [0,\infty)$. The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to $\mathcal{U}$, which depends on the dimension-reduction phenomena.

math.PR↗

arXiv2017

Representations of Max-Stable Processes via Exponential Tilting

The recent contribution Dieker & Mikosch (2015) [1] obtained important representations of max-stable stationary Brown-Resnick random fields $ζ_Z$ with a spectral representation determined by a Gaussian process $Z$. With motivations from \cite{DM} we derive for some general $Z$, representations for $ζ_Z$ via exponential tilting of $Z$. Our main findings concern a) Dieker-Mikosch representations of max-stable processes, b) two-sided extensions of stationary max-stable processes, c) inf-argmax representation of any max-stable distribution, and d) new formulas for generalised Pickands constants. Our applications include new conditions for the stationarity of $ζ_Z$, a characterisation of Gaussian random vectors and an alternative proof of Kabluchko's characterisation of Gaussian processes with stationary increments.

math.PR↗

arXiv2017

Uniform tail approximation of homogenous functionals of Gaussian fields

Let $X(t),t\in R^d$ be a centered Gaussian random field with continuous trajectories and set $ξ_u(t)= X(f(u)t),t\in R^d$ with $f$ some positive function. Classical results establish the tail asymptotics of $P\{ Γ(ξ_u) > u\}$ as $u\to \infty$ with $Γ(ξ_u)= \sup_{t \in [0,T ]^d} ξ_u(t),T>0$ by requiring that $f(u) \to 0$ with speed controlled by the local behaviour of the correlation function of $X$. Recent research shows that for applications more general continuous functionals than supremum should be considered and the Gaussian field can depend also on some additional parameter $τ_u \in K$, say $ξ_{u,τ_u}(t),t\in R^d$. In this contribution we derive uniform approximations of $P\{ Γ(ξ_{u,τ_u})> u\}$ with respect to $τ_u$ in some index set $K_u$, as $u\to\infty$. Our main result have important theoretical implications; two applications are already included in [10,11]. In this paper we present three additional ones, namely i) we derive uniform upper bounds for the probability of double-maxima, ii) we extend Piterbarg-Prisyazhnyuk theorem to some large classes of homogeneous functionals of centered Gaussian fields $ξ_{u}$, and iii) we show the finiteness of generalized Piterbarg constants.

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↗