Pickands Constant Research Articles

Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants

Let $X(t),t\in \mathbb{R}$ be a stochastically continuous stationary max-stable process with Fr\'{e}chet marginals $\Phi_\alpha, \alpha>0$ and set $M_X(T)=\sup_{t \in [0,T]} X(t),T>0$. In the light of the seminal articles [1,2], it follows that $A_T=M_X(T)/T^{1/\alpha}$ converges in distribution as $T\to \infty$ to $\mathcal{H}_Z^{1/\alpha} X(1)$, where $\mathcal{H}_Z$ is the Pickands constant corresponding to the spectral process $Z$ of $X$. In this contribution we derive explicit formulas for $\mathcal{H}_Z$ in terms of $Z$ and show necessary and sufficient conditions for its positivity. From our analysis it follows that $A_T^\beta,T>0$ is uniformly integrable for any $\beta \in (0,\alpha)$. Further, we discuss the dissipative Rosi\'nski (or mixed moving maxima) representation of $X$. Additionally, for Brown-Resnick $X$ we show the validity of the celebrated Slepian inequality and obtain lower bounds on the growth of supremum of Gaussian processes with stationary increments by exploiting the link between Pickands constants and Wills functional. Moreover, we derive upper bounds for supremum of centered Gaussian processes given in terms of Wills functional, and discuss the relation between Pickands and Piterbarg constants.

Approximation of Sojourn Times of Gaussian Processes

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes X, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of Berman (Commun Pure Appl Math 38(5):519–528, 1985a and Probab Theory Relat Fields 20(1):113–124, 1987) for non-stationary X. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.

Remarks on Pickands’ theorem

REMARKS ON PICKANDS’ THEOREMIn this article we present the Pickands theorem and his double sum method. We follow Piterbarg’s proof of this theorem. Since his proof relies on general lemmas, we present a complete proof of Pickands’ theorem using the Borell inequality and Slepian lemma. The original Pickands’ proof is rather complicated and is mixed with upcrossing probabilities for stationary Gaussian processes. We give a lower bound for Pickands constant. Moreover, we review equivalent definitions, simulations and bounds of Pickands constant.

Extremes of [formula omitted]-Ornstein–Uhlenbeck processes

Two limit theorems are established on the extremes of a family of stationary Markov processes, known as q-Ornstein–Uhlenbeck processes with q∈(−1,1). Both results are crucially based on the weak convergence of the tangent process at the lower boundary of the domain of the process, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown–Resnick-type limit theorem on the minimum process of i.i.d. copies of the q-Ornstein–Uhlenbeck process: with appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.

Pickands’ constant at first order in an expansion around Brownian motion

In the theory of extreme values of Gaussian processes, many results are expressed in terms of the Pickands constant . This constant depends on the local self-similarity exponent α of the process, i.e. locally it is a fractional Brownian motion (fBm) of Hurst index . Despite its importance, only two values of the Pickands constant are known: and . Here, we extend the recent perturbative approach to fBm to include drift terms. This allows us to investigate the Pickands constant around standard Brownian motion () and to derive the new exact result .

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 Levy 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.

On Numerical Computation of Pickands Constants

Pickands constant appears in the classical result about tail probabilities of the extremes of Gaussian processes and there exist several different representations of Pickands constant. However, the exact value of is unknown except for two special Gaussian processes. Significant effort has been made to find numerical approximations of . In this paper, we attempt to compute numerically based on its representation derived by (1999) and Albin and Choi (2010). Our estimates are compared with the often quoted conjecture $H_{\alpha}

Open problems in Gaussian fluid queueing theory

We present three challenging open problems that originate from the analysis of the asymptotic behavior of Gaussian fluid queueing models. In particular, we address the problem of characterizing the correlation structure of the stationary buffer content process, the speed of convergence to stationarity, and analysis of an asymptotic constant associated with the stationary buffer content distribution (the so-called Pickands constant).

Ruin probability for Gaussian integrated processes

Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes. By the generalized Pickands constant H η we mean the limit H η= lim T→∞ H η(T) T , where H η(T)= E exp( max t∈[0,T] ( 2 η(t)−σ η 2(t))) and η( t) is a centered Gaussian process with stationary increments and variance function σ η 2( t). Under some mild conditions on σ η 2( t) we prove that H η is well defined and we give a comparison criterion for the generalized Pickands constants. Moreover we prove a theorem that extends result of Pickands for certain stationary Gaussian processes. As an application we obtain the exact asymptotic behavior of ψ(u)= P( sup t⩾0 ζ(t)−ct>u) as u→∞, where ζ(x)= ∫ 0 x Z(s) ds and Z( s) is a stationary centered Gaussian process with covariance function R( t) fulfilling some integrability conditions.

On tail probabilities and first passage times for fractional Brownian motion

In the paper we present a method of simulation of ruin probability over infinite horizon for fractional Brownian motion with parameter of self-similarity H >½. We derive some theoretical results which show how fast the method works. As an application of our method we numerically compute the Pickands constant.