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.

Let q∈ℕ, let G=(0,∞)^q, and let S:G× E⟶ E, (r,x)⟼ S_rx be a jointly measurable left action on an arbitrary measurable space (E,ℰ). For α=(α_1,…,α_q)∈(0,∞)^q set χ_α(r)=∏_i=1^q r_i^α_i. We study nonzero σ-finite measures ν satisfying ν(S_rA)=χ_α(r)^-1ν(A), r∈ G, A∈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) = 𝔼{∫_G𝕀_A(S_rZ)∏_i=1^q α_i r_i^-α_i-1dr_i}, A∈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.

Let X(t), t∈ K, be a centred Gaussian process with continuous sample paths on a compact metric space K, and let M=min_t∈ KX(t). Let σ_*^2 denote the minimum covariance energy associated with X, and assume that σ_*^2>0. Motivated by the results of for smooth Gaussian processes, we show that, conditionally on M>u, the scaled overshoot u(M-u) converges, as u→∞, 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.

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.

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.

This paper investigates random-shift representations of $\alpha$-homogeneous shift-invariant classes of random fields (rf's) $ K_{\alpha}[ Z]$, which were introduced in \cite{hashorva2021shiftinvariant}. Here $ Z(t),t\in T$ is a stochastically continuous $\mathbb{R}^d$-valued rf with $ T=\mathbb{R}^l$ or $T=\mathbb{Z}^l$. We show that that random-shift representations of interest are obtained by constructing cluster rf's, which play a crucial role in the study of extremes of stationary regularly varying rf's. An important implication of those representations is their close relationship with Rosi\'nski (or mixed moving maxima) representations of max-stable rf's. We show that for a given $ K_{\alpha}[ Z]$ different cluster rf's can be constructed, which is useful for the derivation of new representations of extremal functional indices, Rosi\'nski representations of max-stable rf's as well as for random-shift representations of shift-invariant tail measures.

We study shift-generated classes of jointly measurable and separable ℝ^d-valued random fields (RFs) indexed by 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.

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 ℓ_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≤ 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 ℓ_1-Wasserstein formula when p=1, and discuss applications to Archimax and clustered Archimax copulas and to Brown–Resnick/Hüsler–Reiss models.

Given an Rd-valued random field (rf) Z(t),t∈T and an α-homogeneous mapping κ we define the corresponding equivalent class of rf's (denoted by Cκ[Z]) which include representers of the same tail measure νZ. 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 Cκ[Z]'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 Cκ[Z] 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.

Let Z(t)= exp( √( 2) B_H(t)- |t |^2H) , t∈ℝ with B_H(t),t∈ℝ a standard fractional Brownian motion (fBm) with Hurst parameter H ∈ (0,1] and define for x non-negative the Berman function ℬ_Z(x)= 𝔼{𝕀{ϵ _0(RZ) > x}/ϵ _0(RZ)}∈ (0,∞ ), where the random variable R independent of Z has survival function 1/x,x⩾ 1 and ϵ _0(RZ) = ∫ _ℝ𝕀{ RZ(t)> 1}dt . 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 presented in this article.

We study the asymptotics of sojourn time of the stationary queueing process Q(t),t≥ 0 fed by a fractional Brownian motion with Hurst parameter H∈ (0,1) above a high threshold u . For the Brownian motion case H=1/2 , we derive the exact asymptotics of ℙ{∫ _T_1^T_2𝕀(Q(t)>u+h(u))d t>x |Q(0) >u } as u→∞ , where T_1,T_2, x≥ 0 and T_2-T_1>x , whereas for all H∈ (0,1) , we obtain sharp asymptotic approximations of ℙ{1/v(u)∫ _[T_2(u),T_3(u)]𝕀(Q(t)>u+h(u))dt>y |1/v(u)∫ _[0,T_1(u)]𝕀(Q(t)>u)dt>x } , x,y >0 as u→∞ , 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.

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 of 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.

Motivated by the classical harmonic mean formula, estabished by Aldous in 1989, we investigate the relation between the sojourn time and supremum of a random process X(t), t is an element of 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.

Gaussian random processes whose variances reach their maximum values at unique points are considered. Exact asymptotic behavior of probabilities of large absolute maximums of their trajectories have been evaluated using the double sum method under the widest possible conditions.

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.

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.

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

Let $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $\rho \in (-1,1)$ and define the joint survival probability of both supremum functionals $\pi_\rho(c_1,c_2; u, v)$ by $$\pi_\rho(c_1,c_2; u, v)=\mathbb{P}\left(\sup_{s \in [0,1]} \left(W_1(s)-c_1s\right)>u,\sup_{t \in [0,1]} \left(W_2(t)-c_2t\right)>v\right) ,$$ where $c_1,c_2 \in \mathbb{R}$ and $u,v$ are given positive constants. Approximation of $\pi_\rho(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 $\pi_\rho(c_1,c_2; u, v)$ in the case $\rho \in (0,1)$ and obtain precise approximations by letting $u\to \infty$ and taking $v= au$ for some fixed positive constant $a$ and $\rho \in (-1,1).$

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].