Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval

Let $ \mathcal{T} $ be a positive random variable independent of a real-valued stochastic process $ \left\{ {X(t),t\geqslant 0} \right\} $ . In this paper, we investigate the asymptotic behavior of $ \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right) $ as u→∞ assuming that X is a strongly dependent stationary Gaussian process and $ \mathcal{T} $ has a regularly varying survival function at infinity with index λ ∈ [0, 1). Under asymptotic restrictions on the correlation function of the process, we show that $ \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right)={c^{\lambda }}\mathrm{P}\left( {\mathcal{T}>m(u)} \right)\left( {1+o(1)} \right) $ with some positive finite constant c and function m(·) defined in terms of the local behavior of the correlation function and the standard Gaussian distribution.     

Tail asymptotic expansions for L-statistics

We derive higher-order expansions of L-statistics of independent risks X ₁, …,X _n under conditions on the underlying distribution function F. The new results are applied to derive the asymptotic expansions of ratios of two kinds of risk measures, stop-loss premium and excess return on capital, respectively. Several examples and a Monte Carlo simulation study show the efficiency of our novel asymptotic expansions.     

Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Let X(t), t[0,1], be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique fixed point t₀ in the interval [0,1], the behavior of P{sup_t[0,1] X(t)>u} is known for u. We investigate the case where the unique point t₀ = t_u depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes X_u(t) depending on u, which have for each u such a unique point t_u tending to the boundary as u. We derive the asymptotic behavior of P{sup_t[0,1] X(t)>u}, depending on the rate as t_u tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the difficulties to simulate such rare events, when u is large.     

Asymptotic properties of type I elliptical random vectors

Let be an elliptical random vector with a non-singular square matrix and a spherical random vector in , and let be a sequence of vectors in such that . We assume in this paper that the associated random radius R _k =(S ₁ + S ₂ +...+S _k )^1/2 is almost surely positive, and it has distribution function in the Gumbel max-domain of attraction. Relying on extreme value theory we obtain an exact asymptotic expansion of the tail probability for converging as to a boundary point. Further we discuss density convergence under a suitable transformation. We apply our results to obtain an asymptotic approximation of the distribution of partial excess above a high threshold, and to derive a conditional limiting result. Further, we investigate the asymptotic behaviour of concomitants of order statistics, and the tail asymptotics of associated random radius for subvectors of .      

Boundary Non-crossings of Brownian Pillow

Let B ₀(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]²→? be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability

$\psi(u;h):=\mathbf{P}\big\{B_{0}(s,t)+h(s,t)\leq u(s,t),\forall s,t\in[0,1]\big\}.$

Further we investigate the asymptotic behaviour of ψ(u;γ h) with γ tending to ∞ and solve a related minimisation problem.

     

Asymptotics of the norm of elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation , where R is a positive random radius independent of the random vector which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a given matrix. Denote by ‖⋅‖ the Euclidean norm in R^d, and let F be the distribution function of R. The main result of this paper is an asymptotic expansion of the probability for F in the Gumbel or the Weibull max-domain of attraction. In the special case that is a mean zero Gaussian random vector our result coincides with the one derived in Hüsler et al. (2002) [1].     

Generalized Pickands constants and stationary max-stable processes

Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (Bernoulli, 20(3), 1600–1619, 2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker–Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore develop a link to extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.     

Second Order Asymptotics of Aggregated Log-Elliptical Risk

In this paper we establish the error rate of first order asymptotic approximation for the tail probability of sums of log-elliptical risks. Our approach is motivated by extreme value theory which allows us to impose only some weak asymptotic conditions satisfied in particular by log-normal risks. Given the wide range of applications of the log-normal model in finance and insurance our result is of interest for both rare-event simulations and numerical calculations. We present numerical examples which illustrate that the second order approximation derived in this paper significantly improves over the first order approximation.