Extremes of Gaussian random fields with regularly varying dependence structure

Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ ²(?) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t ₀. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t ₀. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t ₀. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.     

Extremes of independent chi-square random vectors

We prove that the componentwise maximum of an i.i.d. triangular array of chi-square random vectors converges in distribution, under appropriate assumptions on the dependence within the vectors and after normalization, to the max-stable Hüsler–Reiss distribution. As a by-product we derive a conditional limit result.     

On the residual dependence index of elliptical distributions

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius has distribution function in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.     

Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals

Let {X(t),t ≥ 0} be a centered Gaussian process and let γ be a non-negative constant. In this paper we study the asymptotics of \(\mathbb {P} \left \{\underset {t\in [0,\mathcal {T}/u^{\gamma }]}\sup X(t)>u\right \}\) as \(u\rightarrow \infty \) , 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.     

Boundary noncrossings of additive Wiener fields∗

Let {W _i (t), t ∈ ?₊}, i = 1, 2, be two Wiener processes, and let W ₃ = {W ₃(t), t ∈ ? ₊ ² } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P _f = P{W ₁(t ₁) + W ₂(t ₂) + W ₃(t) + f(t) ≤ u(t), t ∈ ? ₊ ² }, where f, u : ? ₊ ² → ? are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, P _γf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \) , where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W ₁(t ₁) + W ₂(t ₂) + W ₃(t). It turns out that our approach is also applicable for the additive Brownian pillow.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

Conditional limit results for type I polar distributions

Let (S ₁,S ₂) = (R cos(Θ), R sin(Θ)) be a bivariate random vector with associated random radius R which has distribution function F being further independent of the random angle Θ. In this paper we investigate the asymptotic behaviour of the conditional survivor probability when u approaches the upper endpoint of F. On the density function of Θ we impose a certain local asymptotic behaviour at 0, whereas for F we require that it belongs to the Gumbel max-domain of attraction. The main result of this contribution is an asymptotic expansion of , which is then utilised to construct two estimators for the conditional distribution function . Furthermore, we allow Θ to depend on u.      

Representations of max-stable processes via exponential tilting

The recent contribution (Dieker and Mikosch, 2015) obtained representations of max-stable stationary Brown–Resnick process ${\zeta }_Z\left(t\right),t\in {\mathbb{R}}^d$ with spectral process $Z$ being Gaussian. With motivations from Dieker and Mikosch (2015) we derive for general $Z$, representations for ${\zeta }_Z$ via exponential tilting of $Z$. Our findings concern Dieker–Mikosch representations of max-stable processes, two-sided extensions of stationary max-stable processes, inf-argmax representation of max-stable distributions, and new formulas for generalised Pickands constants. Our applications include conditions for the stationarity of ${\zeta }_Z$, a characterisation of Gaussian distributions and an alternative proof of Kabluchko’s characterisation of Gaussian processes with stationary increments.     

Tail asymptotics under beta random scaling

Let X,Y,B be three independent random variables such that X has the same distribution function as YB. Assume that B is a beta random variable with positive parameters α,β and Y has distribution function H with H(0)=0. In this paper we derive a recursive formula for calculation of H, if the distribution function Hα,β of X is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y, which is closely related to asymptotics of Weyl fractional-order integral operators. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and Hα,β, respectively, and the conditional limiting distribution of bivariate elliptical distributions.