Extremes of vector-valued Gaussian processes

The seminal papers of Pickands (Pickands, 1967; Pickands, 1969) paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian’s Lemma, there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting, obtaining the exact asymptotics of the high exceedance probabilities for both stationary and n on-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and Ornstein–Uhlenbeck process.     

Extremes of threshold-dependent Gaussian processes

In this paper,we are concerned with the asymptotic behavior,as u→∞,of P{sup_t∈|0,T|X_u(t)u},where X_u(t),t∈|0,T|,u0 is a family of centered Gaussian processes with continuous trajectories.A key application of our findings concerns P{sup_t∈|0,T|(X(t)+g(t))u},as u→∞,for X a centered Gaussian process and g some measurable trend function.Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest.     

Extremes of space–time Gaussian processes

Let Z={_Zt(h);h∈R^d,t∈R}$Z=\{Z_t\left(h\right);h\in {\mathbb{R}}^d,t\in \mathbb{R}\}$ be a space–time Gaussian process which is stationary in the time variable t$t$. We study _Mn(h)=_{supt∈[0,n]Zt}(_snh)$M_n\left(h\right)={sup}_{t\in [0,n]}{Z}_t\left(s_{n}h\right)$, the supremum of Z$Z$ taken over t∈[0,n]$t\in [0,n]$ and rescaled by a properly chosen sequence _sn→0$s_n\to 0$. Under appropriate conditions on Z$Z$, we show that for some normalizing sequence _bn→∞$b_n\to \infty$, the process _bn(_Mn−_bn)$b_n(M_n-b_n)$ converges as n→∞$n\to \infty$ to a stationary max-stable process of Brown–Resnick type. Using strong approximation, we derive an analogous result for the empirical process.     

Extremes of Gaussian processes with a smooth random variance

Let ξ(t) be a standard stationary Gaussian process with covariance function r(t), and η(t), another smooth random process. We consider the probabilities of exceedances of ξ(t)η(t) above a high level u occurring in an interval [0,T] with T>0. We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ(t)η(t), which is called the random variance process. We derive also a large deviation result for a general class of conditional Gaussian processes X(t) given a random element Y.     

Extremes of independent stochastic processes: a point process approach

For each n ≥ 1, let \(\{ X_{in}, \quad i \geqslant 1 \}\) be independent copies of a nonnegative continuous stochastic process X _n = (X _n (s)) _s∈S indexed by a compact metric space S. We are interested in the process of partial maxima \(\tilde M_{n}(t,s) =\max \{ X_{in}(s), 1 \leqslant i\leqslant [nt] \},\quad t\geq 0,\ s\in S,\) where the brackets [ ? ] denote the integer part. Under a regular variation condition on the sequence of processes X _n , we prove that the partial maxima process \(\tilde M_{n}\) weakly converges to a superextremal process \(\tilde M\) as \(n\to \infty \). We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.     

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.     

Limit theorems for functionals of two independent Gaussian processes

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in [11] and a pairing technique are employed.     

Extremes of stationary Gaussian storage models

For the stationary storage process {Q(t), t ≥ 0}, with \( Q(t)=\sup _{s\ge t}\left (X(s)-X(t)-c(s-t)^{\beta }\right ),\) where {X(t), t ≥ 0} is a centered Gaussian process with stationary increments, c > 0 and β > 0 is chosen such that Q(t) is finite a.s., we derive exact asymptotics of \(\mathbb {P}\left (\sup _{t\in [0,T_{u}]} Q(t)>u \right )\) and \(\mathbb {P}\left (\inf _{t\in [0,T_{u}]} Q(t)>u \right )\), as \(u\rightarrow \infty \). As a by-product we find conditions under which strong Piterbarg property holds.     

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.     

Extremes of Shepp statistics for Gaussian random walk

Let (ξ _i , i ≥ 1) be a sequence of independent standard normal random variables and let be the corresponding random walk. We study the renormalized Shepp statistic and determine asymptotic expressions for when u,N and T→ ∞ in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of when T,N→ ∞ and present corresponding normalization sequences.      

Discrete and Continuous Time Extremes of Gaussian Processes

Joint distribution of maximums of a Gaussian stationary process in continuous time and in uniform grid on the real axis is studied. When the grid is sufficiently sparse, maxima are asymptotically independent. When the grid is sufficiently tight, the maximums asymptotically coincide. In the boundary case which we call Pickands grid, the limit distribution is non-degenerate. It calculated in terms of a Pickands type constant.AMS 2000 Subject Classification. Primary—60G70, Secondary—60G15*Partially supported by the Scientific foundation of the Netherlands, RFFI grant 0401-00700 and grant DFG 436 RUS 113/722.     

Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes

We derive the asymptotic behavior for an additive functional of two independent self-similar Gaussian processes when their intersection local time exists, using the method of moments.     

Invariant Gaussian processes and independent sets on regular graphs of large girth

We prove that every 3‐regular, n‐vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d‐regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue . We show that such processes can be approximated by i.i.d. factors provided that . We then use these approximations for to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 284–303, 2015     

Asymptotic Poisson Character of Extremes in Non-Stationary Gaussian Models

Let X be a non-stationary Gaussian process, asymptotically centered with constant variance. Let u be a positive real. Define R_u(t) as the number of upcrossings of level u by the process X on the interval (0, t]. Under some conditions we prove that the sequence of point processes (R_u)_u>0 converges weakly, after normalization, to a standard Poisson process as u tends to infinity. In consequence of this study we obtain the weak convergence of the normalized supremum to a Gumbel distribution.AMS 2000 Subject Classifications Primary—60G70, 60G15