arXiv subjects

Enkelejd Hashorva

Publications and source records attributed to Enkelejd Hashorva.

At least 19 recordsLinked to original sources

arXiv2026

Multihomogeneous Measures and Stochastic Polar Representations

Let $q\in\mathbb N$, let $G=(0,\infty)^q$, and let $ S:G\times E\longrightarrow E, (r,x)\longmapsto S_rx $ be a jointly measurable left action on an arbitrary measurable space $(E,\mathcal E)$. For $\alpha=(\alpha_1,\ldots,\alpha_q)\in(0,\infty)^q$ set $ \chi_\alpha(r)=\prod_{i=1}^q r_i^{\alpha_i}. $ We study nonzero $\sigma$-finite measures $\nu$ satisfying $ \nu(S_rA)=\chi_\alpha(r)^{-1}\nu(A), r\in G, A\in\mathcal E.$ Motivated by the scalar case $q=1$ studied in [1] we derive equivalent conditions for the existence of an $E$-valued random element $Z$ such that $ \nu(A) = \mathbb{E}\{\int_G\mathbb I_A(S_rZ)\prod_{i=1}^q \alpha_i r_i^{-\alpha_i-1}dr_i\}, A\in\mathcal E. $ We also characterise when two random elements generate the same homogeneous measure, using multihomogeneous moments and, after fixing an admissible product gauge, weighted transverse measures. When the corresponding weighted transverse measure is finite, tilting and gauge normalisation produce a canonical representer, unique in law on the prescribed gauge shell. Finally, we characterise stationarity under an action commuting with $S$ and construct positive semidefinite tail-overlap kernels directly from $\nu$.

math.PR↗

arXiv2026

Branch-stationary max-stable fields on rooted trees

In this contribution we study max-stable random fields on the rooted tree under shifts to descendant subtrees. Branch-Brown--Resnick stationarity is characterised through homogeneous spectral classes, punctured tail measures, and local spectral tail fields. For lognormal representers, it is equivalent to invariance of the variogram under addition of a common prefix. We give Gaussian, max-autoregressive, regenerative cascade, and free-group cluster constructions, and show that summability on countably branching trees need not satisfy a zero--one law. We also derive the associated branch-invariant extreme-value and Archimax copulas.

math.PR↗

arXiv2026

High Minima of Gaussian Processes: Overshoots and Minimizer Locations

Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $\sigma_*^2$ denote the minimum covariance energy associated with $X$, and assume that $\sigma_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $\sigma_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.

math.PR↗

arXiv2026

Visibility in the Boolean Model on Harmonic Manifolds

In Poisson Boolean models with deterministic ball grains, the directional visible range from an uncovered point is known to be exponentially distributed in Euclidean and real hyperbolic space. We show that the same phenomenon holds on every simply connected non-compact homogeneous harmonic manifold. The geometric mechanism behind this fact is the affine-linear growth of tube volumes around geodesic segments. As a consequence, we identify the finiteness regime for the expected volume of the visible region, including a geometric interpretation of the critical threshold in the positive-entropy case. We also construct explicit complete non-homogeneous Riemannian manifolds showing that exact exponentiality is tied to exact tube linearity: superlinear tube growth leads to Weibull-type tails, while asymptotic tube linearity still yields an exponential decay rate.

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 infimum. We also consider the stationary case and give a new explicit expression for the fidi's of such processes.

math.PR↗

arXiv2025

Kolmogorov and Wasserstein Distances between Max-Stable Distributions

We derive explicit comparison bounds for multivariate max-stable distributions with unit-$\alpha$-Fr\'echet margins. For the Kolmogorov distance, the bounds are expressed through Wasserstein distances between powered de Haan representers, total variation distances between angular measures, and discrepancies of the $\Psi$-functions in the inf--argmax decomposition. On the positive $\ell_\alpha$-sphere, the coefficient multiplying the setwise angular total-variation distance contains no explicit dimension factor for the unnormalised angular measures used here. Separately, for $1\le p<\alpha$, a synchronous de Haan--LePage coupling bounds the $p$-Wasserstein distance between the max-stable laws by an $\alpha$-Wasserstein transport cost between their unpowered de Haan representers. We also compare laws with a common extreme-value copula and different Fr\'echet indices, obtaining an exact $\ell_1$-Wasserstein formula when $p=1$, and discuss applications to Archimax and clustered Archimax copulas and to Brown--Resnick/H\"usler--Reiss models.

math.PR↗

arXiv2025

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 = (b1, ..., bd)^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). To approximate kappa_u(x), we extend both Berman's original approach and the uniform double-sum method to the multivariate setting. Furthermore, we derive the exact asymptotics for the supremum of X, thus extending several recent results in the literature.

math.PR↗

arXiv2025

Shift-generated classes of jointly measurable random fields

We study shift-generated classes of jointly measurable and separable $\mathbb R^d$-valued random fields (RFs) indexed by $\mathbb R^l$, defined through identities for $\alpha$-homogeneous functionals. In contrast to earlier work, no stochastic-continuity assumption and no local boundedness condition are imposed. We show that every non-empty shift-generated class contains an $L^\alpha$-continuous element. This regularization result allows us to establish the strict positivity of the integral functional for all elements of the class and for the associated local RFs. We further extend the defining functional identity to a larger class of functionals, including integral functionals, and use this to construct canonical elements of a given class via randomised shifts. We also relate shift-generated classes to spectral tail and tail RFs and show that every spectral tail RF has an $L^\alpha$-continuous representative with the same finite-dimensional distributions. As an application, we identify the $-\alpha$-homogeneous tail measure associated with a shift-generated class and show that it depends only on the class and admits an $L^\alpha$-continuous representor.

math.PR↗

arXiv2023

Sojourns of fractional Brownian motion queues: transient asymptotics

We study the asymptotics of sojourn time of the stationary queueing process $Q(t),t\ge0$ fed by a fractional Brownian motion with Hurst parameter $H\in(0,1)$ above a high threshold $u$. For the Brownian motion case $H=1/2$, we derive the exact asymptotics of \[ P\left(\int_{T_1}^{T_2} 1(Q(t)>u+h(u))d t>x \Big{|}Q(0) >u \right) \] as $u\to\infty$, {where $T_1,T_2, x\geq 0$ and $T_2-T_1>x$}, whereas for all $H\in(0,1)$, we obtain sharp asymptotic approximations of \[ P\left( \frac 1 {v(u)} \int_{[T_2(u),T_3(u)]}1(Q(t)>u+h(u))dt>y \Bigl \lvert \frac 1 {v(u)} \int_{[0,T_1(u)]}1(Q(t)>u)dt>x\right), \quad x,y >0 \] as $u\to\infty$, for appropriately chosen $T_i$'s and $v$. Two regimes of the ratio between $u$ and $h(u)$, that lead to qualitatively different approximations, are considered.

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↗

arXiv2022

Cluster Random Fields and Random-Shift Representations

Cluster random fields (CRFs) play a crucial role in the study of extremes of stationary regularly varying random fields (RFs). In particular, they appear in the Rosi\'nski representation of max-stable and $\alpha$-stable RFs. In this contribution we introduce CRFs in an abstract setting proving that they are crucial for the construction of shift-generated classes of $\alpha$-homogeneous RFs. Further, we investigate the relations between CRFs, tail RFs} and spectral tail RFs. Applications discussed in this contribution include new representations of extremal functional indices and purely dissipative max-stable RFs.

math.PR↗

arXiv2022

Tail Measures and Regular Variation

A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed RV with respect to some properly localised boundedness $\mathcal{B}$ imposing weak assumptions on the structure of Polish space. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on general Polish metric spaces. Tail measures introduced in [2] appear naturally as limiting measures of regularly varying time series. We define tail measures on a measurable space indexed by $\mathcal{H}(D)$, a countable family of homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $\mathcal{B}$ is determined by $\mathcal{H}(D)$. This allows us to study the regular variation of cadlag processes on $D(R^l, R^d)$ retrieving in particular results obtained in [1] for RV of stationary cadlag processes on the real line removing $l=1$ therein. Further, we discuss potential applications and open questions.

math.PR↗

arXiv2022

The harmonic mean formula for random processes

Motivated by the harmonic mean formula in [1], we investigate the relation between the sojourn time and supremum of a random process $X(t),t\in \mathbb{R}^d$ and extend the harmonic mean formula for general stochastically continuous $X$. We discuss two applications concerning the continuity of distribution of supremum of $X$ and representations of classical Pickands constants.

math.PR↗

arXiv2021

Shift-invariant homogeneous classes of random fields

Given an $R^d$-valued random field (rf) $Z(t),t\in T$ and an $\alpha$-homogeneous mapping $\kappa$ we define the corresponding equivalent class of rf's (denoted by $K_\alpha$) which include representers of the same tail measure $\nu_Z$. When $T$ is an additive group, tractable equivalent classes of interest are the shift-invariant ones, which contain in particular all independent random shifts of $Z$. This contribution is mainly concerned with the investigation of the probabilistic properties of shift-invariant $K_\alpha$'s. Important objects introduced in our setting are tail and spectral tail rf's. Further, the class of universal maps $U$ acting on elements of $K_\alpha$ turns out to be crucial for properties of functionals of $Z$. Applications of our findings concern max-stable and symmetric $\alpha$-stable rf's, their maximal indices as well as their random shift-representations.

math.PR↗

arXiv2021

Pandemic-type Failures in Multivariate Brownian Risk Models

Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical $d$-dimensional Brownian risk model (Brm), see [1]. From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least $k$ out of $d$ components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of [2,3].

math.PR↗

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↗