arXiv subjects

Enkelejd Hashorva

Publications and source records attributed to Enkelejd Hashorva.

At least 19 recordsLinked to original sources

arXiv2026

Multiradial Regular Variation

We introduce and study multiradial regular variation of random fields, allowing componentwise thresholds to diverge at unrelated rates. The limit measures are homogeneous in each component and finite on events where every component exceeds a positive level in absolute value somewhere on a compact window. We characterise convergence by anchor exceedance masses and conditional whole-path laws, together with a compact-window mass bound in the continuous-parameter setting. For fields indexed by a countable discrete abelian group or by R^m every shift-invariant tail measure in this class admits a strictly stationary realisation. Finite moving averages with shared volatility show that scalar row-tail measures and simultaneous-exceedance tail masses on every consecutive finite window can coincide while relative-lag tail masses differ.

math.PR↗

arXiv2026

Expected Infimum and persistence probabilities of Log-Normal Stationary Brown-Resnick Processes

We investigate the asymptotics of the expected infimum of log-normal Brown-Resnick stationary processes, a class of processes that arise naturally in the study of extremes of Gaussian processes and max-stable processes. Specifically, we analyse the functional $$\mathcal{G}_V(T) = \mathbb{E}\left\{\inf_{t \in [0,T]}e^{ \sqrt{2}V(t)-σ^2_V(t)} \right\}, \quad T>0, $$ where $V$ is a centered Gaussian process with stationary increments, continuous sample paths and variance $σ_V^2$, and a closely related problem of the decay rate of the persistence probability $$p_V(T,C)=\mathbb{P}\left\{\inf_{t\in [0,T]} (\sqrt{2} V(t)- σ^2_V(t)) > C\right\}$$ for some constant $C<0$. For both $\mathcal{G}_V(T)$ and $p_V(T,C)$ we derive exact asymptotics as $T \to \infty$ for a broad class of processes $V$, including fractional Brownian motion with Hurst parameter $H \in (1/2,1]$. For the latter, the behavior in the short-range dependence regime $H \in (0, 1/2]$ is markedly different and, in general, more delicate; we find logarithmic asymptotics for a family of processes that includes this case. Our results provide sharp bounds and comparison principles, and highlight the contrast between the behavior of infimum and supremum functionals for such processes. The discrete-time analogues and connections to Pickands constants are also discussed.

math.PR↗

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 $α=(α_1,\ldots,α_q)\in(0,\infty)^q$ set $ χ_α(r)=\prod_{i=1}^q r_i^{α_i}. $ We study nonzero $σ$-finite measures $ν$ satisfying $ ν(S_rA)=χ_α(r)^{-1}ν(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 $ ν(A) = \mathbb{E}\{\int_G\mathbb I_A(S_rZ)\prod_{i=1}^q α_i r_i^{-α_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 $ν$.

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

Exact Tail Asymptotics of Dirichlet Distributions

Let $\X=A^\top R\U$ be a linearly transformed generalised symmetrised Dirichlet scale mixture in $\R^k$, $k\ge2$. For a fixed direction $\b\in(0,\infty)^k$, we derive an exact asymptotic expansion of $\pk{\X>\vk t_n}$ for eventually positive threshold vectors $\vk t_n$ described relative to the quadratic-programming minimiser on the natural active and residual Gumbel scales; residual limits equal to $-\infty$ are allowed. The radial distribution is assumed to belong to the Gumbel max-domain of attraction. The local power and constant are determined by the local product-power behaviour of the angular density near the minimising direction. The result includes the ray $\vk t_n=u_n\b$ and yields an explicit comparison with the associated elliptical model, a conditional weak limit for the locally rescaled vector and the limiting location of the smallest component under a high common threshold. The minimum overshoot is asymptotically exponential and independent of its location. The finite-dimensional Gaussian minimum and location limits are recovered as a special case.

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 $σ_*^2$ denote the minimum covariance energy associated with $X$, and assume that $σ_*^2>0$. 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 $σ_*^2$. 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

Kolmogorov and Wasserstein Distances between Max-Stable Distributions

We derive explicit comparison bounds for multivariate max-stable distributions with unit-$α$-Fréchet 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 $Ψ$-functions in the inf--argmax decomposition. On the positive $\ell_α$-sphere, the coefficient multiplying the setwise angular total-variation distance contains no explicit dimension factor for the unnormalised angular measures used here. Separately, for $1\le p<α$, a synchronous de Haan--LePage coupling bounds the $p$-Wasserstein distance between the max-stable laws by an $α$-Wasserstein transport cost between their unpowered de Haan representers. We also compare laws with a common extreme-value copula and different Fréchet indices, obtaining an exact $\ell_1$-Wasserstein formula when $p=1$, and discuss applications to Archimax and clustered Archimax copulas and to Brown--Resnick/Hüsler--Reiss models.

math.PR↗

arXiv2026

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 $α$-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 $L^α$-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 $L^α$-continuous representative with the same finite-dimensional distributions. As an application, we identify the $-α$-homogeneous tail measure associated with a shift-generated class and show that it depends only on the class and admits an $L^α$-continuous representor.

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

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

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 Rosiński representation of max-stable and $α$-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 $α$-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↗

arXiv2024

Shift-invariant homogeneous classes of random fields

Given an $R^d$-valued random field (rf) $Z(t),t\in T$ and an $α$-homogeneous mapping $κ$ we define the corresponding equivalent class of rf's (denoted by $K_α$) which include representers of the same tail measure $ν_Z$. 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 $K_α$'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 $K_α$ turns out to be crucial for properties of functionals of $Z$. Applications of our findings concern max-stable and symmetric $α$-stable rf's, their maximal indices as well as their random shift-representations.

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

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

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↗